Proof That Homeomorphisms Preserve The Continuum Property
Ifis a homeomorphism andis a continuum, then so is
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.