Кейіндемелік топология және жиын теориясының қатынастары

In mathematics, set theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).

In the mathematical field of general topology, a Dowker space is a topological space 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. Zoltán Balogh gave the first ZFC construction of a small (cardinality continuum) example, which was more well behaved 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.

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. 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 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 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 cellularity of a space X is is a family of mutually disjoint non empty open subsets of The Hereditary cellularity (sometimes spread) is the least upper bound of cellularities of its subsets: or with the subspace topology is discrete 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 .

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 universe V 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.