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A theorem about (or providing an equivalent definition of) compact sets, originally due to Georg Cantor. Given a decreasing sequence of bounded nonempty closed sets C_1 ...

There are several equivalent definitions of a closed set. Let S be a subset of a metric space. A set S is closed if 1. The complement of S is an open set, 2. S is its own set ...

A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.

A topological space X such that for every closed subset C of X and every point x in X\C, there is a continuous function f:X->[0,1] such that f(x)=0 and f(C)={1}. This is the ...

The expression im kleinen is German and means "on a small scale." A topological space is connected im kleinen at a point x if every neighborhood U of x contains an open ...

A general mathematical property obeyed by mathematical objects in which all elements are within a neighborhood of nearby points. The continuous maps between topological ...

A Banach algebra A is called contractible if H^1(A,X)=Z^1(A,X)/B^1(A,X)=0 for all Banach A-bimodules X (Helemskii 1989, 1997). A C^*-algebra is contractible if and only if it ...

A covering map (also called a covering or projection) is a surjective open map f:X->Y that is locally a homeomorphism, meaning that each point in X has a neighborhood that is ...

A derivation is a sequence of steps, logical or computational, from one result to another. The word derivation comes from the word "derive." "Derivation" can also refer to a ...

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space ...