Definition:Homeomorphism/Metric Spaces/Definition 2

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Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $f: A_1 \to A_2$ be a bijection such that:

for all $U \subseteq A_1$, $U$ is an open set of $M_1$ if and only if $f \sqbrk U$ is an open set of $M_2$.


Then:

$f$ is a homeomorphism
$M_1$ and $M_2$ are homeomorphic.


Also known as

A homeomorphism between two metric spaces is also known as a topological equivalence.

Two homeomorphic metric spaces can be described as topologically equivalent.


Also see


Sources