A 1-to-1 mapping between topological spaces that preserves the topological structure, i. e. the open sets. Homeomorphic spaces have essentially the same topological properties.
If ''X'' and ''Y'' are spaces, a homeomorphism from ''X'' to ''Y'' is a bijective function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''−1 are continuous. The spaces ''X'' and ''Y'' are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.
A one-to-one continuous transformation that preserves open and closed sets.
a bijection that is continuous and whose inverse is also continuous
a bijective map of which the inverse is a map, too
an injective (one-to-one), continuous transformation of one topological space onto another whose inverse is also continuous
a one-to-one, continuous m apping of R d onto R d with a continuous www
a one to one, continuous mapping of R with a continuous inverse
homeomorphism from a space to a space is a bijective map : → such that and - are continuous. The spaces and are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.
In graph theory, two graphs G and G′ are homeomorphic if there is an isomorphism from some subdivision of G to some subdivision of G′. If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.
