View source for Zero-dimensional space

''in the sense of $\mathrm{ind}$''

A topological space having a base of sets that are at the same time open and closed in it. Every [[Discrete space|discrete space]] is zero-dimensional, but a zero-dimensional space need not have isolated points (an example is the space $\mathbf{Q}$ of rational numbers). All zero-dimensional spaces are completely regular. Zero-dimensionality is inherited by subspaces and implies [[Totally-disconnected space|total disconnectedness]] of the space: The only connected sets in a zero-dimensional space are the [[singleton]]s and the empty set. However, the latter property is not equivalent to being zero-dimensional. There are spaces that are not zero-dimensional and in which every point is the intersection of a family of open-and-closed sets, but no such space can be compact.

Sometimes the zero-dimensionality of a space is understood more narrowly. A space is called zero-dimensional in the sense of $\mathrm{dim}$ if every finite open covering of it can be refined to an open covering with disjoint elements. A space is called zero-dimensional in the sense of $\mathrm{Ind}$ if any neighbourhood of any closed subset of it contains an open-and-closed neighbourhood of this subset. In the class of [[T1 space|$T_1$-space]]s, zero-dimensionality in the sense of $\mathrm{ind}$ follows from both that in the sense of $\mathrm{dim}$ and that in the sense of $\mathrm{Ind}$. In the class of [[metrizable space]]s with a countable base, and also in the class of Hausdorff compacta, the three definitions of being zero-dimensional are equivalent. For all metrizable spaces, zero-dimensionality in the sense of $\mathrm{dim}$ is equivalent to that in the sense of $\mathrm{Ind}$; however, an example is known of a metrizable space that is zero-dimensional in the sense of $\mathrm{ind}$, but not in the sense of $\mathrm{Ind}$. Neither zero-dimensionality in the sense of $\mathrm{Ind}$ nor that in the sense of $\mathrm{dim}$ is inherited by subspaces. Among $T_1$-spaces the zero-dimensional ones in the sense of $\mathrm{ind}$ can be characterized, up to a homeomorphism, as subspaces of generalized Cantor discontinua $D^\tau$ — products of [[colon]]s. Any [[completely-regular space]] can be obtained as the image of a zero-dimensional space under a well-behaved mapping, for example, under a [[perfect mapping]] and under a continuous [[open mapping]] with compact inverse images of points. However, continuous mappings that are simultaneously open and closed preserve zero-dimensionality in the sense of $\mathrm{ind}$ and of $\mathrm{Ind}$. It is not known whether every completely-regular space contains an everywhere-dense zero-dimensional subspace.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   B.A. Pasynkov,   "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR>
</table>

====Comments====
A set that is simultaneously open and closed is sometimes called a clopen set.

The $T_0$ separation axiom is frequently assumed (without explicit mention) as part of the definition of zero-dimensionality: indeed, some of the assertions in the main article above are false if this assumption is not made.

Zero-dimensionality in the sense of $\mathrm{Ind}$ and $\mathrm{dim}$ are equivalent. This type of zero-dimensionality is generally called strong zero-dimensionality. It is clear that strongly zero-dimensional spaces are normal. A normal space $X$ is strongly zero-dimensional if and only if its [[Stone–Čech compactification]] $\beta X$ is zero-dimensional (in any sense). This fact motivates the extension of the notion of strong zero-dimensionality to the class of completely-regular spaces: A completely-regular space $X$ is defined to be strongly zero-dimensional if its compactification $\beta X$ is zero-dimensional.

Every topological space is an open quotient of a zero-dimensional paracompact space [[#References|[a4]]]. Completely-regular spaces are perfect images even of [[extremally-disconnected space]]s.

A fourth definition of dimension agreeing with the basic three in separable metric spaces is Krull dimension $\mathrm{gdim}$; see [[#References|[a5]]] and (the editorial comments to) [[Lattice]]. For general metric spaces $X$, $\mathrm{ind}(X) \le \mathrm{gdim}(X) \le \mathrm{dim}(X)$ [[#References|[a5]]].

For completely-regular spaces one may define zero-dimensional spaces that map onto it in many ways. One important such construction is that of the [[Absolute|absolute]], or Gleason cover. Initially, A.M. Gleason constructed $\alpha Y$ for a compact Hausdorff space $Y$, and showed that it is the projective covering of $Y$ in a sense dual to that of injective hull (see [[Injective module|Injective module]]) [[#References|[a3]]]. In fact, applied to zero-dimensional compact Hausdorff spaces (also called Boolean spaces or Stone spaces, cf. [[Stone space|Stone space]]), the construction is dual by Stone duality (see [[Boolean algebra|Boolean algebra]]) to the MacNeille completion of Boolean algebras, which is an instance of injective hulls (cf. [[Completion, MacNeille (of a partially ordered set)]]); equipped with appropriate  "compatible"  orderings, they also provide a duality (Priestly duality) for arbitrary distributive lattices. The Gleason cover has been considerably generalized: to $T_0$-spaces [[#References|[a2]]], to topological algebras of general type [[#References|[a1]]], and to toposes [[#References|[a6]]].

The question at the end of the article has been solved partially: Assuming the [[Continuum hypothesis]] one can construct a space such that no uncountable subspace of it is zero-dimensional and such that no countable subset is dense.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Banaschewski,   "Projective covers in categories of topological spaces and topological algebras"  J. Novák (ed.)  et al. (ed.) , ''General topology and its relations to modern analysis and algebra (Proc. Kanpur, 1968)'' , '''3''' , Academia  (1971)  pp. 63–91</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Błaszczyk,   "Extremally disconnected resolutions of $T_0$ spaces"  ''Colloq. Math.'' , '''32'''  (1974)  pp. 57–68</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Gleason,   "Projective topological spaces"  ''Ill. J. Math.'' , '''2'''  (1958)  pp. 482–489</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Isbell,   "A note on complete closure algebras"  ''Math. Systems Theory'' , '''3'''  (1969)  pp. 310–312</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  J.R. Isbell,   "Graduation and dimension in locales"  I.H. James (ed.)  E.H. Kronheimer (ed.) , ''Aspects of Topology: in Memory of Hugh Dowker'' , ''Lect. notes London Math. Soc.'' , '''93''' , Cambridge Univ. Press  (1985)  pp. 195–210</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  P.T. Johnstone,   "The Gleason cover of a topos I"  ''J. Pure Appl. Alg.'' , '''19'''  (1980)  pp. 171–192</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  P.T. Johnstone,   "The Gleason cover of a topos II"  ''J. Pure Appl. Alg.'' , '''22'''  (1981)  pp. 229–247</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  K. Ciesielski,   "$L$-space without any uncountable $0$-dimensional subspace"  ''Fundam. Math.'' , '''125'''  (1985)  pp. 231–235</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  R. Engelking,   "Dimension theory" , North-Holland  (1978)  (Translated from Polish)</TD></TR>
<TR><TD valign="top">[a11]</TD> <TD valign="top">  W. Hurevicz,   G. Wallman,   "Dimension theory" , Princeton Univ. Press  (1948)  ((Appendix by L.S. Pontryagin and L.G. Shnirel'man in Russian edition.))</TD></TR>
<TR><TD valign="top">[a12]</TD> <TD valign="top">  P. Nyikos,   "A survey of zero-dimensional spaces"  S.P. Franklin (ed.)  et al. (ed.) , ''Topology (Proc. 9th Annual Spring Conf. Memphis, 1975)'' , M. Dekker  (1976)  pp. 87–114</TD></TR>
<TR><TD valign="top">[a13]</TD> <TD valign="top">  P.T. Johnstone,   "Stone spaces" , Cambridge Univ. Press  (1983)</TD></TR>
</table>

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