Meaning of topological space | Babel Free

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