topological space

Definitions

1. An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);
   formal
2. An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets); (loosely) the set X.
   broadly, formal
3. the set X.
   broadly, formal

Equivalents

Examples

“In order⟳ to obtain⟳ "intuitive insight" into special classes of topological spaces we can proceed in several ways, only a few of which will be pursued in this chapter. For instance, we can seek⟳ to describe⟳ important topological spaces by means of enough of their properties to completely characterize them, up to homeomorphism.”

“2011, Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, Lecture⟳ Notes in Mathematics 2032, page xi, Most of the spaces studied in Algebraic Topology, such as CW-complexes or manifolds, are Hausdorff. In contrast, finite topological spaces are rarely Hausdorff. A topological space with finitely many points, each of which is closed, must be discrete.”

“If (X,#92;tau) is a topological space, then a cover⟳ #92;mathcalB is open⟳ if each B#92;in#92;mathcalB is an open⟳ set⟳. A topological space (X,#92;tau) is compact if every open⟳ cover⟳ of X has a finite subcover.”

CEFR level