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Topological Space

Alternate name

abstract topological space

Basic definition

A topological space is a set with a collection of subsets T that together satisfy a certain set of axioms defining the topology of that set.

Detailed definition

A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions: 1. The empty set ∅ is in T. 2.X is in T. 3. The intersection of a finite number of sets in T is also in T. 4. The union of an arbitrary number of sets in T is also in T. Alternatively, T may be defined to be the closed sets rather than the open sets, in which case conditions 3 and 4 become: 3. The intersection of an arbitrary number of sets in T is also in T.

Related terms

closed set | Hausdorff axioms | Kuratowski's closure-complement problem | manifold | neighborhood | open neighborhood | open set | sigma-compact topological space | T_2-space | topological vector space | topology

Educational grade level

college level

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