Vipul: New page: ==Definition== ===In terms of the metric=== Suppose $(X,d_X)$ and $(Y,d_Y)$ are fact about::metric spaces. In other words, $X$ and $Y$ are...

2008-11-24T23:04:05Z

New page: ==Definition== ===In terms of the metric=== Suppose <math>(X,d_X)</math> and <math>(Y,d_Y)</math> are fact about::metric spaces. In other words, <math>X</math> and <math>Y</math> are...

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==Definition==

===In terms of the metric===

Suppose <math>(X,d_X)</math> and <math>(Y,d_Y)</math> are [[fact about::metric space]]s. In other words, <math>X</math> and <math>Y</math> are sets, and <math>d_X</math> and <math>d_Y</math> are metrics on these sets. A function <math>f:X \to Y</math> is termed a '''continuous map''' from <math>X</math> to <math>Y</math> if it satisfies the following:

<math>\forall \ \varepsilon > 0, a \in X, \ \exists \ \delta > 0 : d_X(a,b) < \delta \implies d_Y(f(a),f(b)) < \varepsilon</math>.

===In terms of the induced topology===

Suppose <math>(X,d_X)</math> and <math>(Y,d_Y)</math> are metric spaces. A function <math>f:X \to Y</math> is termed a '''continuous map''' if <math>f</math> is a continuous map from <math>X</math> to <math>Y</math> with the [[metric induces topology|induced topologies from their respective metrics]].

The notion of continuous map of metric spaces gives rise to the notion of the [[category of metric spaces with continuous maps]].

Continuous map of metric spaces - Revision history

Vipul: New page: ==Definition== ===In terms of the metric=== Suppose (X,d_X) and (Y,d_Y) are fact about::metric spaces. In other words, X and Y are...

Vipul: New page: ==Definition== ===In terms of the metric=== Suppose $(X,d_X)$ and $(Y,d_Y)$ are fact about::metric spaces. In other words, $X$ and $Y$ are...