Answer to Question #116435 in Differential Geometry | Topology for Sheela John

Question #116435
Prove or disprove any metric defined on X(#0) induces a topology on X
1
Expert's answer
2020-05-24T15:55:59-0400

Every metric space is a topological space in a natural manner.

The metric topology is the topology on "(X,d)" generated by the base "B_r(x)=\\{y\\in X\\colon d(x,y)<r\\}" (open balls). It is base because

  1. Union of "B_r(x)" cover "X" (obviously)
  2. "x\\in B_{r_1}(x_1)\\cap B_{r_2}(x_2)" then "x\\in B_r(x) \\subset B_{r_1}(x_1)\\cap B_{r_2}(x_2)" with "r=\\min\\{r_1-d(x,x_1),r_2-d(x,x_2)\\}"

A base generates a topology on "X" that has, as open sets, all unions of elements of a base.


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