In
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, geometri ...
. 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 ...
(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, geometri ...
, 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 po ...
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 in 1971. Rudin's counterexample is a very large space (of
cardinality ) and is generally not
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
.
Zoltán Balogh gave the first
ZFC construction of a small (cardinality
continuum) example, which was more
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
than Rudin's. Using
PCF theory, M. Kojman and
S. Shelah constructed a
subspace of Rudin's Dowker space of cardinality
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
as a tool for describing various
topological properties
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
. 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 "
" 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
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
w(''X'' ) of a topological space ''X'' is the smallest possible cardinality of a
base for ''X''. When w(''X'' )
the space ''X'' is said to be ''
second countable''.
** The
-weight of a space ''X'' is the smallest cardinality of a
-base for ''X''. (A
-base is a set of nonempty opens whose supersets includes all opens.)
* The
character of a topological space ''X'' at a point ''x'' is the smallest cardinality of a
local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
for ''x''. The character of space ''X'' is
When
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
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 ra ...
of ''X''. When
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
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
has a subcover of cardinality no more than L(''X'' ). When
the space ''X'' is said to be a ''
Lindelöf space''.
* The cellularity of a space ''X'' is
.
** The Hereditary cellularity (sometimes spread) is the least upper bound of cellularities of its subsets:
or
.
* The tightness ''t''(''x'', ''X'') of a topological space ''X'' at a point
is the smallest cardinal number
such that, whenever
for some subset ''Y'' of ''X'', there exists a subset ''Z'' of ''Y'', with , ''Z'' , ≤
, such that
. Symbolically,
The tightness of a space ''X'' is
. When ''t(X) = ''
the space ''X'' is said to be ''
countably generated'' or ''
countably tight''.
** The augmented tightness of a space ''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 ...
such that for any
,
there is a subset ''Z'' of ''Y'' with cardinality less than
, such that
.
Martin's axiom
For any cardinal k, we define a statement, denoted by MA(k):
For any partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
''P'' satisfying the countable chain condition 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.
Overview
There are really two conditions: the ''upwards'' and ''downwards'' countable c ...
(hereafter ccc) and any family ''D'' of dense sets in ''P'' such that '', D, '' ≤ k, there is a filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
''F'' on ''P'' such that ''F'' ∩ ''d'' is non-empty
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
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
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
.
MA(
) 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 o ...
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
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
, which is
separable and so ccc. It has no
isolated point
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
s, so points in it are nowhere dense, but it is the union of
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 po ...
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
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
,
analytic and
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
consequences:
* The union of k or fewer
null sets in an atomless σ-finite
Borel measure on a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named be ...
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
In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X.
Every metric space is naturally a topological space, and for metric spaces, the notio ...
, i.e., every sequence has a convergent subsequence.
* No non-principal
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
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(
) 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
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:
Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory.
Refinement
Assume that ''A'' is an abel ...
is independent of ZFC.
Forcing
Forcing is a technique invented by
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
for proving
consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
and
independence
Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the statu ...
results. It was first used, in 1963, to prove the independence of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
and the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
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 ...
. 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
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
such as
recursion theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has sinc ...
.
Intuitively, forcing consists of expanding the set theoretical
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. ...
''V'' to a larger universe ''V''*. In this bigger universe, for example, one might have many new
subsets of
''ω'' = that were not there in the old universe, and thereby violate the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
. While impossible on the face of it, this is just another version of
Cantor's paradox
In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is ...
about infinity. In principle, one could consider
:
identify
with
, and then introduce an expanded membership relation involving the "new" sets of the form
. 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