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) 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 that is also Dowker.
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 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 "" 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 ) 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 Xat a pointx is the smallest cardinality of a
local base for x. The character of space X is
The density d(X ) of a space X is the smallest cardinality of a
dense subset 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 has a subcover of cardinality no more than L(X ). When the space X is said to be a Lindelöf space.
The tightnesst(x, X) of a topological space Xat 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 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 .
For any cardinal k, we define a statement, denoted by MA(k):
For any
partial orderP satisfying the
countable chain condition (hereafter ccc) and any family D of dense sets in P such that |D| ≤ k, there is a
filterF 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.
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.
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() implies that a product of ccc topological spaces is ccc (this in turn implies there are no
Suslin lines).
Intuitively, forcing consists of expanding the set theoretical
universeV 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
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
^M.E. Rudin, A normal space X for which X × I is not normal, Fundam. Math.73 (1971) 179-186. Zbl. 0224.54019