Set-theoretic topology
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, set-theoretic topology is a subject that combines
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and
general topology 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, geomet ...
. It focuses on topological questions that are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
of
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(ZFC).


Objects studied in set-theoretic topology


Dowker spaces

In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
general topology 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, geomet ...
, a Dowker space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
that is T4 but not
countably paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
. Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until M.E. Rudin constructed one in 1971. Rudin's counterexample is a very large space (of
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
\aleph_\omega^) and is generally not well-behaved. Zoltán Balogh gave the first ZFC construction of a small (cardinality
continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...
) example, which was more well-behaved than Rudin's. Using
PCF theory PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more ap ...
, M. Kojman and S. Shelah constructed a subspace of Rudin's Dowker space of cardinality \aleph_ 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. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology", 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'' ) \le \aleph_0 the space ''X'' is said to be ''
second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
''. ** 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 Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of a topological space ''X'' at a point ''x'' is the smallest cardinality of a local base for ''x''. The character of space ''X'' is
\chi(X)=\sup \; \.
When \chi(X) \le \aleph_0 the space ''X'' is said to be ''
first countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
''. * The density d(''X'' ) of a space ''X'' is the smallest cardinality of a dense subset of ''X''. When \rm(X) \le \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(X) = \aleph_0 the space ''X'' is said to be a ''
Lindelöf space 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'' sub ...
''. * The cellularity of a space ''X'' is
(X)=\sup\.
** The Hereditary cellularity (sometimes spread) is the least upper bound of cellularities of its subsets:
s(X)=(X)=\sup\
or
s(X)=\sup\.
* The tightness ''t''(''x'', ''X'') of a topological space ''X'' at a point x\in X is the smallest cardinal number \alpha such that, whenever x\in_X(Y) for some subset ''Y'' of ''X'', there exists a subset ''Z'' of ''Y'', with , ''Z'' , ≤ \alpha, such that x\in_X(Z). Symbolically,
t(x,X)=\sup\big\.
The tightness of a space ''X'' is t(X)=\sup\. When ''t(X) = ''\aleph_0 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 In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite ...
\alpha such that for any Y\subseteq X, x\in_X(Y) there is a subset ''Z'' of ''Y'' with cardinality less than \alpha, such that x\in_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^) is false:
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
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^ many points. An equivalent formulation is: If ''X'' is a compact Hausdorff
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
which satisfies the ccc then ''X'' is not the union of k or fewer
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
subsets. Martin's axiom has a number of other interesting combinatorial, analytic and topological consequences: * The union of k or fewer
null set In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...
s in an atomless σ-finite
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. F ...
on a Polish space is null. In particular, the union of k or fewer subsets of R of
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
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 Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
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 line In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither ...
s). * MA + ¬CH implies that there exists a Whitehead group that is not free;
Shelah Shelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading * Salih, a prophet described ...
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 In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
. 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
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of ''ω'' = 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 \, \, 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.


References


Further reading

* {{Topology General topology Set theory