

Continuous Maps and Homotopy

When we need to compare two topological spaces, we
need to use continuous maps between them.
For example, we regard two spaces to be the same if
there is a homeomorphism between them.
When we study
covering spaces
and
orbifolds
, we need local properties of maps.
We sometimes need maps having “half” of
propeties of homeomorphisms and homotopy equivalences.
These concepts seem to have been studied in the early
days of topology as “theory of retracts”.
See Borsuk’s book
[
Bor67
]
and/or Hu’s book
[
Hu65
]
.


