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.
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.
The weight w of a topological space X is the smallest possible cardinality of a base for X. When w the spaceX is said to be second countable.
* The -weight of a space X is the smallest cardinality of a -base for X.
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
When the space X is said to be first countable.
The density d 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 of a space X is the smallest infinite cardinalitysuch that every open cover has a subcover of cardinality no more than L. When the space X is said to be a Lindelöf space.
The tightnesst of a topological space Xat a point is the smallest cardinal number such that, whenever for some subsetY of X, there exists a subset Z of Y, with |Z | ≤, such that. Symbolically,
The tightness of a spaceX is. When t = 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.
For any cardinal k, we define a statement, denoted by MA:
For any partial orderP satisfying the countable chain condition 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 fails, the Martin's axiom is stated as:
Martin's axiom : For every k < c, MA holds.
In this case, an antichain is a subset A of P such that any two distinct members of A are incompatible. This differs from, for example, the notion of antichain in the context of trees. MA is false: is a compactHausdorff 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 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 ≥ k, where χ is the character of x, and so χ ≥ k.
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 universeV 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. 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.
