Proof That Homeomorphisms Preserve The Continuum Property

Theorem

Ifis a homeomorphism andis a continuum, then so is

Proof

The continuous image of a compact set is compact and of a connected set is connected.

Also, ifis a homeomorphism andis T2, then so is

Hence, ifandare homeomorphic andis a continuum, then so is

For example, ifis a continuum andis a continuous real valued function, thenis either a single point or a closed interval.