Continuous Mappings Between Metric Spaces

From Department of Mathematics at UTSA
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Types of maps between metric spaces

Suppose and are two metric spaces.

Continuous maps

The map is continuous if it has one (and therefore all) of the following equivalent properties:

General topological continuity
for every open set in , the preimage is open in
This is the general definition of continuity in topology.
Sequential continuity
if is a sequence in that converges to , then the sequence converges to in .
This is sequential continuity, due to Eduard Heine.
ε-δ definition
for every and every there exists such that for all in we have
This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.

Moreover, is continuous if and only if it is continuous on every compact subset of .

The image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected.

Uniformly continuous maps

The map is uniformly continuous if for every there exists such that

Every uniformly continuous map is continuous. The converse is true if is compact (Heine–Cantor theorem).

Uniformly continuous maps turn Cauchy sequences in into Cauchy sequences in . For continuous maps this is generally wrong; for example, a continuous map from the open interval onto the real line turns some Cauchy sequences into unbounded sequences.

Lipschitz-continuous maps and contractions

Given a real number , the map is K-Lipschitz continuous if

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.

If , then is called a contraction. Suppose and is complete. If is a contraction, then admits a unique fixed point (Banach fixed-point theorem). If is compact, the condition can be weakened a bit: admits a unique fixed point if

.

Isometries

The map is an isometry if

Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete, respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed (or open).

Quasi-isometries

The map is a quasi-isometry if there exist constants and such that

and a constant such that every point in has a distance at most from some point in the image .

Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the "large-scale structure" of metric spaces; they find use in geometric group theory in relation to the word metric.

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