Identification map

(See attached file for full problem description with proper symbols and equations) --- • Let be the subspace of of all positive real numbers. Show that the map defined by is an identification map. ---

Continuous and identification maps

(See attached file for full problem description with proper symbols and equation) --- Let be a surjective continuous map between topological spaces. Show that: a) If f is an identification mp, then for any pace Z and any map the composition is continuous if and only if g is continuous. b) If, for any space Z and any map , the composition is continuous if and only if g is continuous, then f is an identification map. ---

Homeomorphism

(See attached file for full problem description with proper symbols) --- Let and a map, given by . Let ~ be the equivalence relation on Xx[0,1] defined by and all other points are equivalent only to themselves. Show that Xx[0,1]/~ is homeomorphic to the Moebius strip. ---

homotopy

(See attached file for full problem description with proper symbols) --- Let X and Y be topological spaces and let be a subspace. Show that 'homotopic relative A' defines an equivalence relation on the set of continuous maps from X to Y which agree with some fixed map on A. ---

Contractible spaces

(See attached file for full problem description with all symbols) --- Let X be a contractible space: a) Show that X is path connected b) Show that any two continuous maps where Y is any topological space, are homotopic. c) Let and be the map defined by . Show that and are homotopic. ---