Theorem
Ifis a homeomorphism and
is 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 and
is T2, then so is
Hence, ifand
are homeomorphic and
is a continuum, then so is
For example, ifis a continuum and
is a continuous real valued function, then
is either a single point or a closed interval.