View source for Cardinal characteristic

{{MSC|54A25}}

''of a [[topological space]]''

A function associating an infinite [[cardinal number]] to each space and taking the same value on [[Homeomorphism|homeomorphic]] spaces. Cardinal characteristics are also called cardinal invariants. The domain of definition of a cardinal invariant is the class of all topological spaces or some subclass of it. The following cardinal invariants arose at the first stage of development of general topology. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203601.png" /> be an arbitrary topological space. A trivial invariant is its [[cardinality]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203602.png" />, i.e. the cardinality of the set of all its points. Its [[Weight of a topological space|weight]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203603.png" /> is the smallest cardinality of a [[base]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203604.png" />. The [[Density (of a topological space)|density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203605.png" /> is the smallest cardinality of a dense subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203606.png" />. The [[Suslin number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203607.png" /> is the least infinite cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203608.png" /> such that the cardinality of every family of pairwise-disjoint non-empty open sets does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c0203609.png" />. The [[Lindelöf number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036010.png" /> is the least infinite cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036011.png" /> such that every open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036012.png" /> has a subcovering of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036013.png" />. These simple notions immediately showed their importance by entering in a decisive way in fundamental theorems and problems. Examples: a regular space of countable weight is metrizable (the Urysohn–Tikhonov theorem, 1925); a compact Hausdorff space is metrizable if and only if its weight is countable; the Suslin number of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036014.png" /> of a compact group is countable; for every space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036015.png" /> of countable weight its Lindelöf number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036016.png" /> is countable. The Suslin problem — is it true that every ordered connected compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036017.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036018.png" /> is homeomorphic to the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036019.png" /> — leads to the question of the relationship between two cardinal invariants: the density and the Suslin number. For a positive solution of Suslin's problem it is sufficient to show, under the above assumptions, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036020.png" />. The question of comparison of cardinal invariants — the solution of which, as is clear from the above example, may be of key significance for a definitive conclusion on the structure of a space — is central in the theory of cardinal invariants. The reason for this lies in the very nature of the concept of a cardinal invariant: the values of a cardinal invariant are cardinal numbers, the class of which is well-ordered by magnitude. Consequently, one can try to compare the values of any two cardinal invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036022.png" />. A series of mutually related questions arises. Is it true that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036024.png" />; for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036025.png" /> does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036026.png" /> hold; when is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036027.png" />, etc.

It is possible to do arithmetic with cardinals: to multiply and to add them, and to raise them to a power. Correspondingly, it is possible to do arithmetic with cardinal invariants — to multiply and to add them as functions, etc. This opens up new possibilities for comparing cardinal invariants, using arithmetic. There always holds

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036028.png" /></td> </tr></table>

that is, the Suslin number does not exceed the density, the density does not exceed the weight, and the Lindelöf number does not exceed the weight. But the density and the Lindelöf number are not comparable in this sense: There are spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036030.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036031.png" /></td> </tr></table>

The incomparability of cardinality and weight is unexpected at first sight: There are countable normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036032.png" />-spaces of uncountable weight. But always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036034.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036035.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036037.png" /> (one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036038.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036039.png" />). For every Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036041.png" />. Always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036042.png" />.

In a comparison problem it may happen that not just two, but more cardinal characteristics are involved. In that direction, particularly subtle, beautiful and often unexpected results have been obtained, striking in their generality: For each Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036045.png" /> is the least infinite cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036046.png" /> such that at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036047.png" /> there is a local base of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036048.png" /> (see [[#References|[1]]], [[#References|[2]]]). Much research in the theory of cardinal invariants was stimulated by the problem of estimating the cardinality of a compact Hausdorff space satisfying the first axiom of countability, a problem which remained unsolved from 1923 to 1969. It then turned out that for each Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036050.png" /> (Arkhangel'skii's theorem, see [[#References|[2]]], [[#References|[4]]]).

The computation of cardinal invariants takes place in all parts of general topology because of the set-theoretic nature of the latter. Therefore, the theory of cardinal invariants finds application in practically all domains of general topology and in each approach to the investigation of spaces.

In particular, in the study of spaces by coverings, the Lindelöf number, the density and the Suslin number appeared from the very beginning. In the investigation and classification of spaces by continuous mappings (in particular, in the development of the theory of dyadic compacta and of the theory of absolutes) new cardinal invariants arose and played a key role: the spread and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036051.png" />-weight. The spread <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036052.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036053.png" /> is the least upper bound of the cardinalities of discrete subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036054.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036056.png" />-weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036057.png" /> of a space is the minimum of the cardinalities of families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036058.png" /> (called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036060.png" />-bases) of non-empty open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036061.png" /> such that for each non-empty open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036063.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036064.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036065.png" />. In the investigation of spaces by inverse spectra a major role is played by the Suslin number, the character and the weight.

Thus, there is an approach to general topology for which cardinal invariants appear both as the principal means of investigation of the structure of spaces, as a basic language in which the properties of spaces from various classes can be expressed and, finally, as a means of classification and selection of new classes of topological spaces. Basic here is, again, the problem of the comparison of cardinal characteristics. The fundamental question can be posed as follows. Given a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036066.png" /> of topological spaces to which the domains of definition of cardinal invariants are restricted. What are the basic relations between the cardinal invariants under these restrictions? By developing the theory of cardinal invariants for a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036067.png" /> one obtains the  "cardinal portrait"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036068.png" />. A comparison of the cardinal portraits of two classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036070.png" /> allows one to judge the relationships between these classes and also to give effective means of proving that a concrete space belongs to one class or other.

This approach can be demonstrated by the class of metrizable spaces. The characteristic feature here is that for this class a number of fundamental cardinal invariants coincide: the Suslin number is equal to the density, to the weight and to the Lindelöf number. This fact is often applied; for example, to prove that some space is non-metrizable, it is enough to prove that at least two of the invariants mentioned above differ.

In the class of metrizable spaces the theory of cardinal invariants distinguishes itself from the general theory mainly by its simplifications, whereas in the class of compact Hausdorff spaces it changes its appearance completely and in a non-trivial way. Responsible for the particular appearance of this theory is the fact that for compact Hausdorff spaces the character and pseudo-character coincide, as well as the weight and the [[network weight]]. The pseudo-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036072.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036073.png" /> is the smallest number of open sets whose intersection is the point, and the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036074.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036075.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036076.png" /> is the least cardinality of a local base at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036077.png" />. The network weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036078.png" /> is the least cardinality of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036079.png" /> of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036080.png" /> satisfying the condition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036082.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036083.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036084.png" />, then there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036085.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036086.png" /> (such families are called networks in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036087.png" />). For every compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036088.png" /> the following hold: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036089.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036090.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036091.png" />. Therefore, the weight cannot increase under a continuous mapping onto a compact Hausdorff space, and if a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036092.png" /> is the union of two subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036094.png" />, then the weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036095.png" /> does not exceed the maximum of the weights of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036097.png" /> (the addition theorem for weights). For the same reason, the weight of a compact Hausdorff space never exceeds its cardinality; in particular, every countable compact Hausdorff space is metrizable. None of these theorems of the theory of cardinal invariants for the class of compact Hausdorff spaces can be extended to the class of completely-regular spaces. An important specific result is the following: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036098.png" /> is a compact Hausdorff space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c02036099.png" /> is a cardinal number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360100.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360101.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360102.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360103.png" /> (the Čech–Pospišil theorem). Almost-all metrizability criteria for compact Hausdorff spaces are also theorems about cardinal invariants. Thus, metrizability of a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360104.png" /> is equivalent to any of the following conditions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360105.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360106.png" />; c) the diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360107.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360108.png" />-set; or d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360109.png" /> has a point-countable base.

In the investigation of the structure of compact Hausdorff spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360110.png" /> the tightness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360111.png" /> plays a major role. The tightness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360112.png" /> (see [[#References|[2]]], [[#References|[4]]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360113.png" /> is the least cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360114.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360117.png" />, then there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360118.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360119.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360120.png" />. The tightness does not increase when a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360121.png" /> is raised to a finite power (in the class of completely-regular spaces this is not true).

If the tightness of a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360122.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360123.png" />, then for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360124.png" /> there is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360125.png" /> of non-empty open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360126.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360127.png" /> and each neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360128.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360129.png" /> contains an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360130.png" />. Therefore the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360131.png" />-weight of each separable compact Hausdorff space of countable tightness is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360132.png" />. The spread of a compact Hausdorff space majorizes its tightness.

The fundamental properties of dyadic compacta are also, to a large extent, governed by theorems on cardinal characteristics. Thus, for each dyadic compactum the weight coincides with the spread and the tightness. The class of dyadic compacta contains the class of compact Hausdorff topological groups, thus, in particular, every compact Hausdorff group of countable tightness is metrizable.

In the theory of dyadic compacta (cf. [[Dyadic compactum|Dyadic compactum]]) and in other parts of the theory of cardinal invariants, the question of the behaviour of these invariants under multiplication is of major importance. The following two theorems play an essential role here; the first of these implies the second. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360133.png" /> is a family of spaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360134.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360135.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360136.png" />, then the density of the product of the spaces from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360137.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360138.png" /> (see [[#References|[1]]]–[[#References|[4]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360139.png" /> is the product of any set of spaces of densities not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360140.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360141.png" />. In the latter result there is no limit on the number of factors. In particular, the Suslin number of any Tikhonov cube (the product of an arbitrary set of segments) is countable. Thus, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360142.png" /> places no restriction on the cardinality of a space.

Many simply formulated questions on the behaviour of cardinal invariants under multiplication have turned out to be very delicate. For example, the question: Is it true that always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360143.png" />? turns out to be related to the Suslin conjecture and the continuum hypothesis.

On the other hand, the behaviour of cardinal invariants when passing from a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360144.png" /> to its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360145.png" /> under a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360146.png" />, is, on the whole, governed by simple general rules.

For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360147.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360148.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360150.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360151.png" /> is a quotient mapping onto, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360152.png" />. The fact that the foundation of the theory of cardinal invariants consists of a system of simple universal rules of this kind also can be considered as one of the reasons ensuring the broad applicability of the theory.

Significant information on the structure of spaces is obtained by consideration of the question: How do cardinal invariants behave on passing to a subspace? A cardinal invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360153.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360154.png" /> always implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360155.png" /> is called monotone. These include: weight, network weight, tightness, character, and spread. Non-monotone are the Suslin number, the density and the Lindelöf number. The following questions arise: Which are the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360156.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360157.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360158.png" />; which are the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360159.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360160.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360161.png" />; what effect on the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360162.png" /> has the requirement: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360163.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360164.png" />? The answer to the first question is simple: the condition means that the spread of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360165.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360166.png" />. But the two subsequent conditions single out new classes of spaces. The investigation of these classes turns out to depend on special hypotheses of set theory, in particular on Martin's axiom.

The theory of cardinal invariants has a peculiar character on the spaces of topological groups. E.g., the criterion for metrizability reduces here simply to the first axiom of countability. The major properties of linear topological spaces, in particular, of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360167.png" /> of continuous real-valued functions on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360168.png" />, can be formulated in the language of cardinal invariants. This refers to the theorems on Eberlein compacta (each Eberlein compactum is a Fréchet–Urysohn space, the weight of an Eberlein compactum is equal to its Suslin number); and the following theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360169.png" /> is a compact Hausdorff space, then the tightness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360170.png" /> in the topology of pointwise convergence is countable.

Between a number of cardinal invariants of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360171.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360172.png" /> there is a duality-type correspondence.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Juhász,   "Cardinal functions in topology" , North-Holland  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Engelking,   "Outline of general topology" , North-Holland  (1968)  (Translated from Polish)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Arkhangel'skii,   "Structure and classification of topological spaces, and cardinal invariants"  ''Russian Math. Surveys'' , '''33''' :  6  (1978)  pp. 33–96  ''Uspekhi Mat. Nauk'' , '''33''' :  6  (1978)  pp. 29–84</TD></TR></table>



====Comments====
The usual terminology in the literature (cf. [[#References|[1]]]–[[#References|[4]]]) is cardinal function, or cardinal invariant.

The Suslin number of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360173.png" /> is also called its cellularity, and its Lindelöf number also its Lindelöf degree (the latter is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360174.png" />).

The Urysohn–Tikhonov theorem, mentioned above, is also called Urysohn's metrization theorem.

The fact that the class of dyadic compacta contains the class of compact Hausdorff topological groups is called Kuzminov's theorem (on compact groups).

For the notion of a Fréchet–Urysohn space see [[Sequential space|Sequential space]].

The problem of whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360175.png" /> for every space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360176.png" /> was solved by S. Todorčević [[#References|[a2]]], who found, without using extra set-theoretic hypotheses, spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360177.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360178.png" />.

The problems whether every hereditarily separable space is Lindelöf and whether every hereditarily Lindelöf space is separable generated a lot of research. Many examples were constructed using various extra set-theoretical assumptions, in particular the [[Continuum hypothesis|continuum hypothesis]]. Todorcevi<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020360/c020360179.png" /> [[#References|[a1]]] showed that the statement  "every hereditarily separable space is Lindelöf"  is consistent with the usual axioms of set theory. For much more information and other recent developments see various articles in [[#References|[a3]]], in particular Chapts. 1; 2, and [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Todorcevic,   "Forcing positive partition relations"  ''Trans. Amer. Math. Soc.'' , '''280'''  (1983)  pp. 703–720</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Todorcevic,   "Remarks on cellularity in products"  ''Compos. Math.'' , '''57'''  (1986)  pp. 357–372</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of set-theoretic topology'' , North-Holland  (1984)  pp. Chapts. 1–2</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Juhász,   "Cardinal functions. Ten years later" , ''MC Tracts'' , '''123''' , North-Holland  (1980)</TD></TR></table>