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 set andis 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.