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A family of subsets of a topological space such that every point has a neighborhood which intersects only a finite number of them.

Let Y^X be the set of continuous mappings f:X->Y. Then the topological space for Y^X supplied with a compact-open topology is called a mapping space.

A continuous map f:X->Y between topological spaces is said to be null-homotopic if it is homotopic to a constant map. If a space X has the property that id_X, the identity ...

A topological space X has a one-point compactification if and only if it is locally compact. To see a part of this, assume Y is compact, y in Y, X=Y\{y} and x in X. Let C be ...

A collection of open sets of a topological space whose union contains a given subset. For example, an open cover of the real line, with respect to the Euclidean topology, is ...

In a topological space X, an open neighborhood of a point x is an open set containing x. A set containing an open neighborhood is simply called a neighborhood.

A partial algebra is a pair A=(A,(f_i^A)_(i in I)), where for each i in I, there are an ordinal number alpha_i and a set X_i subset= A^(alpha_i) such that f_i^A is a function ...

An iterated fibration of Eilenberg-Mac lane spaces. Every topological space has this homotopy type.

A Cartesian product equipped with a "product topology" is called a product space (or product topological space, or direct product).

The topology induced by a topological space X on a subset S. The open sets of S are the intersections S intersection U, where U is an open set of X. For example, in the ...