Ivan Shishkin, Birch Grove

Topological space

Definition / Topology / Usable

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Usable. This concept is clear enough to use, but has not yet been reviewed by another trusted user.

A topological space is a set XX together with a topology τ\tau, that is, a collection of subsets of XX, called the open sets, satisfying the following axioms:

  1. The empty set and the whole space are open: ∅∈τ\emptyset\in \tau and X∈τX\in \tau.
  2. Arbitrary unions of open sets are open: if {Ui}i∈I\{U_{i}\}_{i\in I} is any family of sets in τ\tau, then ⋃i∈IUi∈τ\bigcup_{i \in I} U_i \in \tau.
  3. Finite intersections of open sets are open: if U1,…,Un∈τU_1, \ldots, U_n \in \tau, then U1∩⋯∩Un∈τU_1 \cap \cdots \cap U_n \in \tau.

Practice this concept with exercises

  • Soit (E,d)(E,d) un espace métrique. On note
    T={U⊂E  ;  ∀x∈U, ∃r>0 tel que B(x,r)⊂U}.\mathcal T = \bigl\{ U\subset E \; ; \; \forall x\in U,\ \exists r>0 \text{ tel que } B(x,r)\subset U \bigr\}.Autrement dit, T\mathcal T est l’ensemble des ouverts de (E,d)(E,d).
    Montrer que T\mathcal T est une topologie sur EE

    Open exercise
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