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In [[mathematics]], '''set-theoretic topology''' is a subject that combines [[set theory]] and [[general topology]]. It focuses on topological questions that are [[independence (mathematical logic)|independent]] of [[Zermelo–Fraenkel set theory]](ZFC).

==Objects studied in set-theoretic topology==
===Dowker spaces===
In the [[mathematics|mathematical]] field of [[general topology]], a '''Dowker space''' is a [[topological space]] that is [[normal space|T<sub>4]]</sub> but not [[paracompact space|countably paracompact]].

Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until [[Mary Ellen Rudin|M.E. Rudin]] constructed one<ref>M.E. Rudin, A normal space ''X'' for which ''X &times; I'' is not normal, ''[[Fundamenta Mathematicae|Fundam. Math.]]'' '''73''' (1971) 179-186.  Zbl. 0224.54019</ref> in 1971.  Rudin's counterexample is a very large space (of [[cardinality]] <math>\aleph_\omega^{\aleph_0}</math>) and is generally not [[well-behaved]]. [[Zoltán Tibor Balogh|Zoltán Balogh]] gave the first [[ZFC]] construction<ref>Z. Balogh, [http://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03610-6/S0002-9939-96-03610-6.pdf "A small Dowker space in ZFC"], ''[[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]]'' '''124''' (1996) 2555-2560. Zbl. 0876.54016</ref> of a small (cardinality [[Cardinality of the continuum|continuum]]) example, which was more [[well-behaved]] than Rudin's. Using [[PCF theory]], M. Kojman and [[Saharon Shelah|S. Shelah]] constructed<ref>M. Kojman, S. Shelah: [http://www.ams.org/proc/1998-126-08/S0002-9939-98-04884-9/S0002-9939-98-04884-9.pdf "A ZFC Dowker space in <math>\aleph_{\omega+1}</math>: an application of PCF theory to topology"], ''Proc. Amer. Math. Soc.'', '''126'''(1998), 2459-2465.</ref> a subspace of Rudin's Dowker space of cardinality <math>\aleph_{\omega+1}</math> that is also Dowker.

===Normal Moore spaces===
{{Main|Normal Moore space conjecture}}
A famous problem is the [[Normal_Moore_space_conjecture|normal Moore space question]], a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

===Cardinal functions===

Cardinal functions are widely used in [[topology]] as a tool for describing various [[topological properties]].<ref>{{cite book | last=Juhász | first=István | title=Cardinal functions in topology | publisher=Math. Centre Tracts, Amsterdam | year=1979 | isbn=90-6196-062-2 | url=http://oai.cwi.nl/oai/asset/13055/13055A.pdf}}</ref><ref>{{cite book | last=Juhász | first=István | title=Cardinal functions in topology - ten years later | publisher=Math. Centre Tracts, Amsterdam | year=1980 | isbn=90-6196-196-3 | url=http://oai.cwi.nl/oai/asset/12982/12982A.pdf}}</ref> Below are some examples.  (Note: some authors, arguing that "there are no finite cardinal numbers in general topology",<ref>{{cite book | last = Engelking | first = Ryszard | authorlink=Ryszard Engelking | title=General Topology | publisher=Heldermann Verlag, Berlin | year=1989 | isbn=3885380064 | note=Revised and completed edition, Sigma Series in Pure Mathematics, Vol. 6}}</ref> prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, e.g. by adding "<math>\;\; + \;\aleph_0</math>" to the right-hand side of the definitions, etc.)

* Perhaps the simplest cardinal invariants of a topological space ''X'' are its cardinality and the cardinality of its topology, denoted respectively by |''X''&thinsp;| and ''o''(''X'').
* The '''[[Base (topology)#Weight and character|weight]]''' w(''X''&thinsp;) of a topological space ''X'' is the cardinality of the smallest [[Base (topology)|base]] for ''X''.  When w(''X''&thinsp;) = <math>\aleph_0</math> the space ''X'' is said to be ''[[second countable]]''.
** The '''<math>\pi</math>-weight''' of a space ''X'' is the cardinality of the smallest <math>\pi</math>-base for ''X''.
* The '''[[Base (topology)#Weight and character|character]]''' of a topological space ''X'' '''at a point''' ''x'' is the cardinality of the smallest [[Neighbourhood system|local base]] for ''x''.  The '''character''' of space ''X'' is <center><math>\chi(X)=\sup \; \{\chi(x,X) : x\in X\}.</math></center>  When <math>\chi(X) = \aleph_0</math> the space ''X'' is said to be ''[[First-countable space|first countable]]''.
* The '''density''' d(''X''&thinsp;) of a space ''X'' is the cardinality of the smallest dense subset of ''X''.  When <math>\rm{d}(X) = \aleph_0</math> the space ''X'' is said to be ''[[Separable space|separable]]''.
* The '''[[Lindelöf space#Generalisation|Lindelöf number]]''' L(''X''&thinsp;) of a space ''X'' is the smallest infinite cardinality such that every [[open cover]] has a subcover of cardinality no more than L(''X''&thinsp;). When <math>\rm{L}(X) = \aleph_0</math> the space ''X'' is said to be a ''[[Lindelöf space]]''.
* The '''cellularity''' of a space ''X'' is <center><math>{\rm c}(X)=\sup\{|{\mathcal U}|:{\mathcal U}</math> is a [[family of sets|family]] of mutually [[disjoint sets|disjoint]] non-empty [[open set|open]] subsets of <math>X \}</math>.</center>
** The '''Hereditary cellularity''' (sometimes '''spread''') is the least upper bound of cellularities of its subsets: <center><math>s(X)={\rm hc}(X)=\sup\{ {\rm c} (Y) : Y\subseteq X \}</math></center> or <center><math>s(X)=\sup\{|Y|:Y\subseteq X </math> with the [[subspace (topology)|subspace]] topology is [[discrete topological space|discrete]] <math>\}</math>.</center>
* The '''tightness''' ''t''(''x'', ''X'') of a topological space ''X'' '''at a point''' <math>x\in X</math> is the smallest cardinal number <math>\alpha</math> such that, whenever <math>x\in{\rm cl}_X(Y)</math> for some subset ''Y'' of ''X'', there exists a subset ''Z'' of ''Y'', with |''Z''&thinsp;| ≤ <math>\alpha</math>, such that <math>x\in{\rm cl}_X(Z)</math>.  Symbolically, <center><math>t(x,X)=\sup\big\{\min\{|Z|:Z\subseteq Y\ \wedge\ x\in {\rm cl}_X(Z)\}:Y\subseteq X\ \wedge\ x\in {\rm cl}_X(Y)\big\}.</math></center>  The '''tightness of a space''' ''X'' is <math>t(X)=\sup\{t(x,X):x\in X\}</math>.  When ''t(X) = ''<math>\aleph_0</math> the space ''X'' is said to be ''[[countably generated space|countably generated]]'' or ''[[countable tightness|countably tight]]''.
** The '''augmented tightness''' of a space ''X'', <math>t^+(X)</math> is the smallest [[regular cardinal]] <math>\alpha</math> such that for any <math>Y\subseteq X</math>, <math>x\in{\rm cl}_X(Y)</math> there is a subset ''Z'' of ''Y'' with cardinality less than <math>\alpha</math>, such that <math>x\in{\rm cl}_X(Z)</math>.

===Martin's axiom===
For any cardinal '''k''', we define a statement, denoted by MA('''k'''):

<blockquote>For any [[partial order]] ''P'' satisfying the [[countable chain condition]] (hereafter ccc) and any family ''D'' of dense sets in ''P'' such that ''|D|'' ≤ '''k''', there is a [[filter (mathematics)|filter]] ''F'' on ''P'' such that ''F'' ∩ ''d'' is non-[[empty set|empty]] for every ''d'' in ''D''.</blockquote>

Since it is a theorem of ZFC that MA('''c''') fails, the Martin's axiom is stated as:

<blockquote>'''Martin's axiom (MA):''' For every '''k''' < '''c''', MA('''k''') holds.</blockquote> 

In this case (for application of ccc), an antichain is a subset ''A'' of ''P'' such that any two distinct members of ''A'' are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order). This differs from, for example, the notion of antichain in the context of [[tree (set theory)|trees]]. 

MA(<math>2^{\aleph_0}</math>) is false: [0, 1] is a [[compact space|compact]] [[Hausdorff space]], which is [[separable space|separable]] and so ccc. It has no [[isolated point]]s, so points in it are nowhere dense, but it is the union of <math>2^{\aleph_0}</math> many points.

An equivalent formulation is: If ''X'' is a compact Hausdorff [[topological space]] which satisfies the ccc then ''X'' is not the union of '''k''' or fewer [[nowhere dense]] subsets.

Martin's axiom has a number of other interesting [[combinatorial]], [[Mathematical analysis|analytic]] and [[topological]] consequences:

* The union of '''k''' or fewer [[null set]]s in an atomless σ-finite [[Borel measure]] on a [[Polish space]] is null. In particular, the union of '''k''' or fewer subsets of '''R''' of [[Lebesgue measure]] 0 also has Lebesgue measure 0.
* A compact Hausdorff space ''X'' with ''|X|'' < 2<sup>'''k'''</sup> is  [[Compact space|sequentially compact]], i.e., every sequence has a convergent subsequence.
* No non-principal [[ultrafilter]] on '''N''' has a base of cardinality < '''k'''.
* Equivalently for any ''x'' in β'''N'''\'''N''' we have χ(''x'') ≥ '''k''', where χ is the [[character (topology)|character]] of ''x'', and so χ(β'''N''') ≥ '''k'''.
* MA(<math>\aleph_1</math>) implies that a product of ccc topological spaces is ccc (this in turn implies there are no [[Suslin line]]s).
* MA + ¬CH implies that there exists a [[Whitehead group]] that is not free; [[Saharon Shelah|Shelah]] used this to show that the [[Whitehead problem]] is independent of ZFC.

===Forcing===
{{Main|Forcing (mathematics)}}
'''Forcing''' is a technique invented by [[Paul Cohen (mathematician)|Paul Cohen]] for proving [[consistency]] and [[independence (mathematical logic)|independence]] results. It was first used, in 1963, to prove the independence of the [[axiom of choice]] and the [[continuum hypothesis]] from [[Zermelo–Fraenkel set theory]]. Forcing was considerably reworked and simplified in the 1960s, and has proven to be an extremely powerful technique both within set theory and in areas of [[mathematical logic]] such as [[recursion theory]].

Intuitively, forcing consists of expanding the set theoretical [[universe (mathematics)|universe]] ''V'' to a larger universe ''V''*. In this bigger universe, for example, one might have lots of new [[subset]]s of [[Aleph number#Aleph-naught|''ω'']] = {0,1,2,…} that were not there in the old universe, and thereby violate the [[continuum hypothesis]]. While impossible on the face of it, this is just another version of [[Cantor's paradox]] about infinity. In principle, one could consider 
:<math>V^* = V \times \{0,1\}, \, </math>

identify <math>x \in V</math>with <math>(x,0)</math>, and then introduce an expanded membership relation involving the "new" sets of the form <math>(x,1)</math>. Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set, and allowing for fine control over the properties of the expanded universe.

See the main articles for applications such as random reals.

==References==
{{reflist}}
*{{cite book|title=Handbook of Set-Theoretic Topology|last=Kunen|first=Kenneth|authorlink=Kenneth Kunen|coauthors=Vaughan, Jerry E. (''editors'')|publisher=North-Holland|isbn=0-444-86580-2}}

[[Category:Set theory]]
[[Category:General topology]]