View source for Pseudo-character of a set

''<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" />''

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 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" />.

====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</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></table>