# Set-theoretic topology

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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 independent of Zermelo–Fraenkel set theory (ZFC).

## 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 ${\displaystyle \aleph _{\omega }^{\aleph _{0}}}$) 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 ${\displaystyle \aleph _{\omega +1}}$ 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 "${\displaystyle \;\;+\;\aleph _{0}}$" 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) ${\displaystyle \leq \aleph _{0}}$ the space X is said to be second countable .
• The ${\displaystyle \pi }$-weight of a space X is the smallest cardinality of a ${\displaystyle \pi }$-base for X. (A ${\displaystyle \pi }$-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
${\displaystyle \chi (X)=\sup \;\{\chi (x,X):x\in X\}.}$
When ${\displaystyle \chi (X)\leq \aleph _{0}}$ 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 ${\displaystyle {\rm {{d}(X)\leq \aleph _{0}}}}$ 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 ${\displaystyle {\rm {{L}(X)=\aleph _{0}}}}$ the space X is said to be a Lindelöf space .
• The cellularity of a space X is
${\displaystyle {\rm {c}}(X)=\sup\{|{\mathcal {U}}|:{\mathcal {U}}}$ is a family of mutually disjoint non-empty open subsets of ${\displaystyle X\}}$.
• The Hereditary cellularity (sometimes spread) is the least upper bound of cellularities of its subsets:
${\displaystyle s(X)={\rm {hc}}(X)=\sup\{{\rm {c}}(Y):Y\subseteq X\}}$
or
${\displaystyle s(X)=\sup\{|Y|:Y\subseteq X}$ with the subspace topology is discrete ${\displaystyle \}}$.
• The tightnesst(x, X) of a topological space Xat a point${\displaystyle x\in X}$ is the smallest cardinal number ${\displaystyle \alpha }$ such that, whenever ${\displaystyle x\in {\rm {cl}}_{X}(Y)}$ for some subset Y of X, there exists a subset Z of Y, with |Z| ≤ ${\displaystyle \alpha }$, such that ${\displaystyle x\in {\rm {cl}}_{X}(Z)}$. Symbolically,
${\displaystyle t(x,X)=\sup {\big \{}\min\{|Z|:Z\subseteq Y\ \wedge \ x\in {\rm {cl}}_{X}(Z)\}:Y\subseteq X\ \wedge \ x\in {\rm {cl}}_{X}(Y){\big \}}.}$
The tightness of a spaceX is ${\displaystyle t(X)=\sup\{t(x,X):x\in X\}}$. When t(X) = ${\displaystyle \aleph _{0}}$ the space X is said to be countably generated or countably tight .
• The augmented tightness of a space X, ${\displaystyle t^{+}(X)}$ is the smallest regular cardinal ${\displaystyle \alpha }$ such that for any ${\displaystyle Y\subseteq X}$, ${\displaystyle x\in {\rm {cl}}_{X}(Y)}$ there is a subset Z of Y with cardinality less than ${\displaystyle \alpha }$, such that ${\displaystyle x\in {\rm {cl}}_{X}(Z)}$.

### 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 Fd 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(${\displaystyle 2^{\aleph _{0}}}$) 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 ${\displaystyle 2^{\aleph _{0}}}$ 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| < 2k 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(${\displaystyle \aleph _{1}}$) 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

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

identify ${\displaystyle x\in V}$ with ${\displaystyle (x,0)}$, and then introduce an expanded membership relation involving the "new" sets of the form ${\displaystyle (x,1)}$. 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, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.

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.

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 which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

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 a closed manifold.

In mathematics, a base or basis for the topology τ of a topological space (X, τ) is a family B of open subsets of X such that every open set of the topology is equal to a union of some sub-family of B. 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 mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph.

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 functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable.

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, , behave roughly like . 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 topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.

In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also known as the derived set of .

In set theory, Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All these cardinals are greater than or equal to , the smallest uncountable cardinal, and they are bounded above by , the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets.

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 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.

## References

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