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| − | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756201.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756202.png" />'' | + | {{TEX|done}} |
| | + | ''$A$ in a topological space $X$'' |
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| − | The smallest infinite cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756203.png" /> for which there exists a family of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756204.png" /> of sets open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756205.png" /> with intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756206.png" />. It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756207.png" />. The pseudo-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756208.png" /> is defined for all subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p0756209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562010.png" /> only when all singleton subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562011.png" /> are closed. The pseudo-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562012.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562013.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562014.png" /> is understood to be the pseudo-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562015.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562017.png" />. | + | The smallest infinite cardinal number $\tau$ for which there exists a family of cardinality $\tau$ of sets open in $X$ with intersection $A$. It is usually denoted by $\psi(A,X)$. The pseudo-character $\psi(A,X)$ is defined for all subsets $A$ of $X$ only when all singleton subsets in $X$ are closed. The pseudo-character $\psi(x,X)$ of a point $x\in X$ in a topological space $X$ is understood to be the pseudo-character $\psi(\{x\},X)$ of the set $\{x\}$ in $X$. |
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| − | The pseudo-character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562018.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562019.png" /> is the smallest infinite cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562020.png" /> such that each point is the intersection of a family of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562021.png" /> of sets which are open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562022.png" />. Spaces with countable pseudo-character are those in which every point is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562023.png" />-set (cf. [[Set of type F sigma(G delta)|Set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562024.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562025.png" />)]]). Each topological space can be represented as the image under a continuous open mapping of a paracompact Hausdorff space with a countable pseudo-character. For compact Hausdorff spaces the countability of the pseudo-character is equivalent to the [[First axiom of countability|first axiom of countability]]. In general, the pseudo-character of a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562026.png" /> in a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562027.png" /> is equal to the minimum cardinality of a [[Defining system of neighbourhoods|defining system of neighbourhoods]] of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075620/p07562029.png" />. | + | The pseudo-character $\psi(X)$ of a topological space $X$ is the smallest infinite cardinal number $\tau$ such that each point is the intersection of a family of cardinality $\leq\tau$ of sets which are open in $X$. Spaces with countable pseudo-character are those in which every point is a $G_\delta$-set (cf. [[Set of type F sigma(G delta)|Set of type $F_\sigma$ ($G_\delta$)]]). Each topological space can be represented as the image under a continuous open mapping of a paracompact Hausdorff space with a countable pseudo-character. For compact Hausdorff spaces the countability of the pseudo-character is equivalent to the [[First axiom of countability|first axiom of countability]]. In general, the pseudo-character of a closed set $A$ in a compact Hausdorff space $X$ is equal to the minimum cardinality of a [[Defining system of neighbourhoods|defining system of neighbourhoods]] of the set $A$ in $X$. |
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| | ====References==== | | ====References==== |
Latest revision as of 13:09, 27 September 2014
$A$ in a topological space $X$
The smallest infinite cardinal number $\tau$ for which there exists a family of cardinality $\tau$ of sets open in $X$ with intersection $A$. It is usually denoted by $\psi(A,X)$. The pseudo-character $\psi(A,X)$ is defined for all subsets $A$ of $X$ only when all singleton subsets in $X$ are closed. The pseudo-character $\psi(x,X)$ of a point $x\in X$ in a topological space $X$ is understood to be the pseudo-character $\psi(\{x\},X)$ of the set $\{x\}$ in $X$.
The pseudo-character $\psi(X)$ of a topological space $X$ is the smallest infinite cardinal number $\tau$ such that each point is the intersection of a family of cardinality $\leq\tau$ of sets which are open in $X$. Spaces with countable pseudo-character are those in which every point is a $G_\delta$-set (cf. Set of type $F_\sigma$ ($G_\delta$)). Each topological space can be represented as the image under a continuous open mapping of a paracompact Hausdorff space with a countable pseudo-character. For compact Hausdorff spaces the countability of the pseudo-character is equivalent to the first axiom of countability. In general, the pseudo-character of a closed set $A$ in a compact Hausdorff space $X$ is equal to the minimum cardinality of a defining system of neighbourhoods of the set $A$ in $X$.
References
| [1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |
| [2] | A.V. Arkhangel'skii, "Classes of topological groups" Russian Math. Surveys , 36 : 3 (1981) pp. 151–174 Uspekhi Mat. Nauk , 36 : 3 (1981) pp. 127–146 |
References
| [a1] | R. Engelking, "General topology" , Heldermann (1989) |
| [a2] | K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of set-theoretic topology , North-Holland (1984) pp. Chapts. 1–2 |
How to Cite This Entry:
Pseudo-character of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-character_of_a_set&oldid=33418
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article