Isometry
Suppose
and are metric spaces. Suppose that is a bijection such that Then is called an isometry between and . It preserves the distance between points. And we say that and are isometric.
Homeomorphism
Suppose
and are topological spaces. If is a bijection and both and are continuous we say that is a homeomorphism and that and are homeomorphic.
e.g.
and are not homeomorphic because is compact but is not. and are not homeomorphic. Otherwise, if is a homeomorphism, then it also gives a homeomorphism from . This is impossible, because is not connected while is connected.
Proposition
If
is a homeomorphism then is open in if and only if is open in .
Proof
Topological Property
If some property
of a metric space is such that if has property then so does every metric space that is homeomorphic to we say that is a topological property. More colloquially, these are properties that are only concerned with set-theoretic notions and/or open sets, rather than distances. e.g. Topological properties:
is open in . is finite; countably infinite; or uncountable. has a point such that is open in (an ‘isolated point’) - every continuous real-valued function on
is bounded. Properties that are not topological: is bounded.