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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**

- Let
*f*be the identity map from (**R**^{2},*d*_{2}) to (**R**^{2},*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**R**^{2}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_{}.

- 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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