Metric and Topological Spaces

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## Continuity for topological spaces

As previewed earlier whan we considered open sets in a metric space, we can now make the definition:

Definition
A map f: X Y between topological spaces is continuous if f -1(B) X whenever B Y.

Remark
Note that a continuous map f: X Y "induces" a map from Y to X by B f -1(B).

Definition
A map f: X Y between topological spaces is a homeomorphism or topological isomorphism if f is a continuous bijection whose inverse map f-1 is also continuous.

Remark
By the remark above, such a homeomorphism induces a one-one correspondence between X and Y.

Examples

1. Let f be the identity map from (R2, d2) to (R2, d). Then f is a homeomorphism.
Proof
Since every open set is a union of open neighbourhoods, it is enough to prove that the inverse image of an -neighbourhood is open. This -neighbourhood is an open square in R2 which is open in the usual metric.
A similar proof shows that the image of an -neighbourhood in the usual metric (an open disc) is open in d .

2. In general, if X is a set with two topologies 1 and 2 then the identity map (X, 1) (X, 2) is continuous if 1 is stronger (contains more open sets) than 2 .

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JOC February 2004