Such a map has the property that
It follows that any statement about a topological space which is
ultimately expressible solely in terms of the open sets (together with
set-theoretic relations and operations) will be true for both 
and 
if it is true for either. In other words,
and are indistinguishable as topological spaces. The reader who has had abstract algebra will note
that homeomorphism is the analogy in the setting of topological spaces and continuous functions to the notion of isomorphism
in the setting of groups (or rings) and homomorphisms, and to that of linear isomorphism in the
context of vector spaces and linear maps.
Example
For every space , the identity mapping 
is a homeomorphism.
A property of topological spaces which when possessed by a space is also
possessed by every space homeomorphic to it is called a topological
invariant. We shall meet some examples of such properties later.
One can readily verify that if f is a homeomorphism, then the inverse
mapping 
is also a homeomorphism and that the composition 
of two homeomorphisms f and g is again a homeomorphism. Thus, the relation
`X and Y are homeomorphic' is an equivalence relation.
In general, it may be quite difficult to demonstrate that two spaces are
homeomorphic (unless a homeomorphism is obvious or can easily be
discovered).
For example, to verify that 
is homeomorphic to (0,1) with
its induced metric topology, it is necessary to demonstrate, for instance,
that 
where 
is a homeomorphism.
It is often easier to show that two spaces are not homeomorphic: simply
exhibit an invariant which is possessed by one space and not the other.
Example
The spaces 
and 
are not homeomorphic
since, for example, has the topological invariant `each
nhd is open' while does not.