A function is said to be homeomorphism (topological mapping) if and only if the following conditions are satisfied:
(1) is bijective.
(2)  is continuous.
(3)  is continuous.

It may be noted that if  is a homeomorphism from  to , then  is said to be homeomorphic to  and is denoted by . Form the definition of a homeomorphism, it follows that  and  are homeomorphic spaces, then their points and open sets are put into one-to-one correspondence. In other words,  and  differ only in the nature of their points, but from the point of view of the subject of topology, they are identical or have the same topological structure.

Remarks: : “Homeomorphism” helps reducing the complicated problems in simple form, that is, an apparently complicated space may possibly be homeomorphic to some space more familiar to us. Hence in this way, one determines the properties of complicated spaces easily.

Theorems:

• Bijective continuous mapping  is open if and only if  is continuous.
• If  and  are topological spaces, let  means that  and  are homeomorphic. Then this relation is reflexive, symmetric and transitive.
• Let  and  be topological spaces and  be a bijective function, then the following are equivalent. (1)  is a homeomorphism. (2) For any subset  of ,  is open in  if and only if  is open in . (3) For any subset  of ,  is closed in  if and only if  is closed in . (4) For any subset  of , .