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The collection τ is called a topology on X. The elements of X are usually called points, though they can be any mathematical objects. The sets in τ are called the open sets, and their complements in X are called closed sets. A subset of X may be neither closed nor open, either closed or open, or both.
A function between topological spaces is called
continuous if the inverse image of every open set is open.
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Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a
topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing.
同胚（homeomorphism）
In the mathematical field of topology, a
homeomorphism or topological isomorphism or bicontinuous function is a continuous function
between two topological spaces that has a continuous inverse function. Homeomorphisms are the
isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called
homeomorphic, and from a topological viewpoint they are the same.
同胚（homeomorphism）
同胚（homeomorphism）