Set-theoretic topology

Last updated January 21, 2025

$The space of integers has cardinality 0 {\displaystyle \aleph _{0}} , while the real numbers has cardinality 2 0 {\displaystyle 2^{\aleph _{0}}} . The topologies of both spaces have cardinality 2 0 {\displaystyle 2^{\aleph _{0}}} . These are examples of cardinal functions, a topic in set-theoretic topology. Number-line.svg$

The space of integers has cardinality

\aleph _{0}

, while the real numbers has cardinality

2^{\aleph _{0}}

. The topologies of both spaces have cardinality

2^{\aleph _{0}}

. These are examples of cardinal functions, a topic in set-theoretic topology.

In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that can be solved using set-theoretic methods, for example, Suslin's problem.

Objects studied in set-theoretic topology

Dowker spaces

In the mathematical field of general topology, a Dowker space is a topological space that is T₄ but not countably paracompact.

Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until M.E. Rudin constructed one^[1] in 1971. Rudin's counterexample is a very large space (of cardinality $\aleph _{\omega }^{\aleph _{0}}$ ) and is generally not well-behaved. Zoltán Balogh gave the first ZFC construction^[2] of a small (cardinality continuum) example, which was more well-behaved than Rudin's. Using PCF theory, M. Kojman and S. Shelah constructed^[3] a subspace of Rudin's Dowker space of cardinality $\aleph _{\omega +1}$ that is also Dowker.

Normal Moore spaces

A famous problem is the 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.^[4]^[5] Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology",^[6] prefer to define the cardinal functions listed below so that they never take on finite cardinal numbers as values; this requires modifying some of the definitions given below, e.g. by adding " $\;\;+\;\aleph _{0}$ " 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| and o(X).
The weight w(X ) of a topological space X is the smallest possible cardinality of a base for X. When w(X ) $\leq \aleph _{0}$ $Set-theoretic topology$ the space X is said to be second countable .
- The $\pi$ -weight of a space X is the smallest cardinality of a $\pi$ -base for X. (A $\pi$ -base is a set of nonempty opens whose supersets includes all opens.)
The character of a topological space Xat a pointx is the smallest cardinality of a local base for x. The character of space X is
$\chi (X)=\sup \;\{\chi (x,X):x\in X\}.$
When $\chi (X)\leq \aleph _{0}$ the space X is said to be first countable .
The density d(X ) of a space X is the smallest cardinality of a dense subset of X. When ${\rm {{d}(X)\leq \aleph _{0}}}$ the space X is said to be separable .
The 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 ${\rm {{L}(X)=\aleph _{0}}}$ the space X is said to be a Lindelöf space .
The cellularity of a space X is
${\rm {c}}(X)=\sup\{|{\mathcal {U}}|:{\mathcal {U}}$ is a family of mutually disjoint non-empty open subsets of $X\}$ .
- The Hereditary cellularity (sometimes spread) is the least upper bound of cellularities of its subsets:
  $s(X)={\rm {hc}}(X)=\sup\{{\rm {c}}(Y):Y\subseteq X\}$
  or
  $s(X)=\sup\{|Y|:Y\subseteq X$ with the subspace topology is discrete $\}$ .
The tightnesst(x, X) of a topological space Xat a point $x\in X$ $Set-theoretic topology$ is the smallest cardinal number $\alpha$ $Set-theoretic topology$ such that, whenever $x\in {\rm {cl}}_{X}(Y)$ $Set-theoretic topology$ for some subset Y of X, there exists a subset Z of Y, with |Z | ≤ $\alpha$ $Set-theoretic topology$ , such that $x\in {\rm {cl}}_{X}(Z)$ $Set-theoretic topology$ . Symbolically,
$t(x,X)=\sup {\big \{}\min\{|Z|:Z\subseteq Y\ \wedge \ x\in {\rm {cl}}_{X}(Z)\}:Y\subseteq X\ \wedge \ x\in {\rm {cl}}_{X}(Y){\big \}}.$
The tightness of a spaceX is $t(X)=\sup\{t(x,X):x\in X\}$ $Set-theoretic topology$ . When t(X) = $\aleph _{0}$ $Set-theoretic topology$ the space X is said to be countably generated or countably tight .
- The augmented tightness of a space X, $t^{+}(X)$ is the smallest regular cardinal $\alpha$ such that for any $Y\subseteq X$ , $x\in {\rm {cl}}_{X}(Y)$ there is a subset Z of Y with cardinality less than $\alpha$ , such that $x\in {\rm {cl}}_{X}(Z)$ .

Martin's axiom

For any cardinal k, we define a statement, denoted by MA(k):

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 F on P such that F ∩ d is non-empty for every d in D.

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

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

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 trees.

MA( $2^{\aleph _{0}}$ ) is false: [0, 1] is a compact Hausdorff space, which is separable and so ccc. It has no isolated points, so points in it are nowhere dense, but it is the union of $2^{\aleph _{0}}$ 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, analytic and topological consequences:

The union of k or fewer null sets 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^k is 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 βN\N we have χ(x) ≥ k, where χ is the character of x, and so χ(βN) ≥ k.
MA( $\aleph _{1}$ ) implies that a product of ccc topological spaces is ccc (this in turn implies there are no Suslin lines).
MA + ¬CH implies that there exists a Whitehead group that is not free; Shelah used this to show that the Whitehead problem is independent of ZFC.

Forcing

Forcing is a technique invented by Paul Cohen for proving consistency and 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 V to a larger universe V*. In this bigger universe, for example, one might have many new subsets of 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

V^{*}=V\times \{0,1\},\,

identify $x\in V$ with $(x,0)$ , and then introduce an expanded membership relation involving the "new" sets of the form $(x,1)$ . 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.

Related Research Articles

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence $of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.$

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. For a list of terms specific to algebraic topology, see Glossary of algebraic topology.

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces.

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with closed manifold.

In mathematics, a base (or basis; pl.: bases) for the topology $τ$ of a topological space $(X, τ)$ is a family $of open subsets of X such that every open set of the topology is equal to the union of some sub-family of . For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.$

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe $to a larger universe by introducing a new "generic" object .$

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, 𝔠, behave roughly like ℵ₀. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments.

In functional and convex analysis, and related disciplines of mathematics, the polar set $is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in .$

In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded.

In the mathematical field of general topology, a Dowker space is a topological space that is T₄ but not countably paracompact. They are named after Clifford Hugh Dowker.

In mathematics, a cardinal function is a function that returns cardinal numbers.

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it. Formally, $is dense in if the smallest closed subset of containing is itself.$

In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between , and the cardinality of the continuum, that is, the cardinality of the set $of all real numbers. The latter cardinal is denoted or . A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistent configurations of them.$

In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.

In the mathematical field of set theory, an ultrafilter on a set $is a maximal filter on the set In other words, it is a collection of subsets of that satisfies the definition of a filter on and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of that is also a filter. Equivalently, an ultrafilter on the set can also be characterized as a filter on with the property that for every subset of either or its complement belongs to the ultrafilter.$

References

↑ M.E. Rudin, A normal space X for which X × I is not normal, Fundam. Math. 73 (1971) 179-186. Zbl. 0224.54019
↑ Z. Balogh, "A small Dowker space in ZFC", Proc. Amer. Math. Soc. 124 (1996) 2555-2560. Zbl. 0876.54016
↑ M. Kojman, S. Shelah: "A ZFC Dowker space in $\aleph _{\omega +1}$ : an application of PCF theory to topology", Proc. Amer. Math. Soc., 126(1998), 2459-2465.
↑ Juhász, István (1979). Cardinal functions in topology (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-062-2.
↑ Juhász, István (1980). Cardinal functions in topology - ten years later (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-196-3.
↑ Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3885380064.

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