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.

• homeomorphism
• relative homeomorphism
• homotopy equivalence
• simple homotopy equivalence
• contraction and contractibility of a space

When we study covering spaces and orbifolds , we need local properties of maps.

• local homeomorphism

We sometimes need maps having “half” of propeties of homeomorphisms and homotopy equivalences.

• closed map
• retract
• A is a deformation retract of X
• A is a strong deformation retraction of X
• a space X dominates another space Y

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 ] .

References