Equivalence of Conditions for a Bijection to Be a Homeomorphism

Theorem

Letbe a bijection from a setto a setThe following properties are equivalent.

1.is a homeomorphism.

2.for every

3.is continuous and open (open so that the image of every set is an open set).

4.is continuous and closed (closed so that this image of a closed set is a closed set).

Proof

2. is proved here.

3.is a homeomorphism henceis continuous. Hence, for each open setandis open in

Suppose now thatis continuous and open, then the image of each open set inis an open set inhenceis continuous andis a homeomorphism.

4. Is equivalent to 3. Ifis bijective then the conditions 'is open' and 'is closed' are equivalent. Supposeis open andis closed thenis open and

Hence sinceis open,is closed andis closed.