Equivalence of Conditions for a Bijection to Be a Homeomorphism
Letbe a bijection from a setto a setThe following properties are equivalent.
1.is a homeomorphism.
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).
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.