Topology

compact spaces and path connectedness (see attachment). --- • Show that if is a homeomorphism between topological spaces, then X is path connected if and only if Y is path connected. Using open cover definition: 1) is a compact subset? 2) Is a compact subset? ---

path connecteds

(See attached file for full problem description with symbols) --- • Let be the product space of the path connected topological space . Prove that the product space X is also path connected. ---

Compact spaces and finite sets

(See attached file for full problem description) --- • Prove that a set X with discrete topology is a compact topological space if and only if X is a finite set. ---

Product space and compactnesss

(See attached file for full problem description) --- • Let X and Y be two topological spaces Show that the product space XxY is compact if and only if X and Y are compact. ---

Hausdorff

(See attached file for full problem description) --- • Let X and Y be two topological spaces Show that the product space XxY is Hausdorff space if and only if X and Y are Hausdorff spaces. ---