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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).
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| ==Objects studied in set-theoretic topology==
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| ===Dowker spaces===
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| 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]].
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| 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 × 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.
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| ===Normal Moore spaces===
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| {{Main|Normal Moore space conjecture}}
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| 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.
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| ===Cardinal functions===
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| 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.)
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| * Perhaps the simplest cardinal invariants of a topological space ''X'' are its cardinality and the cardinality of its topology, denoted respectively by |''X'' | and ''o''(''X'').
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| * The '''[[Base (topology)#Weight and character|weight]]''' w(''X'' ) of a topological space ''X'' is the cardinality of the smallest [[Base (topology)|base]] for ''X''. When w(''X'' ) = <math>\aleph_0</math> the space ''X'' is said to be ''[[second countable]]''.
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| ** The '''<math>\pi</math>-weight''' of a space ''X'' is the cardinality of the smallest <math>\pi</math>-base for ''X''.
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| * 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]]''.
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| * The '''density''' d(''X'' ) 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]]''.
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| * The '''[[Lindelöf space#Generalisation|Lindelöf number]]''' L(''X'' ) of a space ''X'' is the smallest infinite cardinality such that every [[open cover]] has a subcover of cardinality no more than L(''X'' ). When <math>\rm{L}(X) = \aleph_0</math> the space ''X'' is said to be a ''[[Lindelöf space]]''.
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| * 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>
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| ** 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>
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| * 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'' | ≤ <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]]''.
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| ** 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>.
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| ===Martin's axiom===
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| For any cardinal '''k''', we define a statement, denoted by MA('''k'''):
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| <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>
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| Since it is a theorem of ZFC that MA('''c''') fails, the Martin's axiom is stated as:
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| <blockquote>'''Martin's axiom (MA):''' For every '''k''' < '''c''', MA('''k''') holds.</blockquote> | |
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| 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]].
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| 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.
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| 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.
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| Martin's axiom has a number of other interesting [[combinatorial]], [[Mathematical analysis|analytic]] and [[topological]] consequences:
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| * 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.
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| * A compact Hausdorff space ''X'' with ''|X|'' < 2<sup>'''k'''</sup> is [[Compact space|sequentially compact]], i.e., every sequence has a convergent subsequence.
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| * No non-principal [[ultrafilter]] on '''N''' has a base of cardinality < '''k'''.
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| * 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'''.
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| * 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).
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| * 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.
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| ===Forcing===
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| {{Main|Forcing (mathematics)}}
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| '''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]].
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| 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
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| :<math>V^* = V \times \{0,1\}, \, </math>
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| 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.
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| See the main articles for applications such as random reals.
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| ==References==
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| {{reflist}}
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| *{{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}}
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| [[Category:Set theory]]
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| [[Category:General topology]]
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