A set together with a collection of subsets, called open sets that satisfy certain axioms. The open sets endow the space with a concept of "nearness" between any two points. This is a generalization of the concept of nearness obtained from a numerical measure of "distance" between two points.

(mathematics) any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional"

a Cantor space if and only if it is non-empty, perfect, compact, totally disconnected, and metrizable

a Cantor space if it is homeomorphic to the Cantor set

a collection of points and a structure that endows them with some coherence, in the sense that we may speak of nearby points or points that in some sense are close together

a pair of sets (X, t ) satisfying the above

a set Sets are one of the most important and fundamental concepts in modern mathematics

a set together with a set whose elements are subsets of , such that If for all , then If and , then Elements of are called open sets of

a set X together with a collection of subsets OS the members of which are called open , with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open

a set x together with a collection t of subsets of x satisfying the following

a space in with a concept of neighborhood

a world where there is no notion of distance or angle, but where there is an idea of continuous motion

A set together with a topology Ï„ that satisfy the topology axioms. Proof by Contradiction

topological space is a set equipped with a collection of subsets of satisfying the following conditions: The empty set and are in . The union of any collection of sets in is also in . The intersection of any pair of sets in is also in . The collection is called a topology on .

Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology.