
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, set-theoretic topology is a subject that combines
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
and
general topology
In mathematics, general topology (or point set 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 differ ...
. It focuses on topological questions that can be solved using set-theoretic methods, for example,
Suslin's problem 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 neith ...
.
Objects studied in set-theoretic topology
Dowker spaces
In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
general topology
In mathematics, general topology (or point set 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 differ ...
, a Dowker space is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
) 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 or n ...
.
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 or n ...
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
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition.
Some sta ...
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 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 family 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_\alpha\su ...
has a subcover of cardinality no more than L(''X'' ). When
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
.
** 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 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 partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
''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
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, e.g., including only woody plants with secondary growth, only p ...
.
MA(
) is false:
, 1
The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
, which is
separable and so ccc. It has no
isolated point
In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton i ...
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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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 as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
,
analytic and
topological
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
consequences:
* The union of k or fewer
null set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers 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 notio ...
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.
...
on a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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 Mathematics, 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 element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...
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 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, 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 awarded a F ...
for proving
consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
and
independence
Independence is a condition of a 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 status of ...
results. It was first used, in 1963, to prove the independence of the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
and the
continuum hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Or equivalently:
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
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 suc ...
. 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 Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
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 since ex ...
.
Intuitively, forcing consists of expanding the set theoretical
universe
The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
''V'' to a larger universe ''V''*. In this bigger universe, for example, one might have many new
subset
In mathematics, a 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 a ...
s of
''ω'' = that were not there in the old universe, and thereby violate the
continuum hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Or equivalently:
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
. While impossible on the face of it, this is just another version of
Cantor's paradox 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