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Continuous functions

A function f: X → Y {\displaystyle f:X\to Y} between topological spaces is called continuous if for every x ∈ X {\displaystyle x\in X} and every neighbourhood N {\displaystyle N} of f (x) {\displaystyle f(x)} there is a neighbourhood M {\displaystyle M} of x {\displaystyle x} such that f (M) ⊆ N. {\displaystyle f(M)\subseteq N.} This relates easily to the usual definition in analysis. Equivalently, f {\displaystyle f} is continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.

In category theory, one of the fundamental categories is Top, which denotes the category of topological spaces whose objects are topological spaces and whose morphisms are continuous functions. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homology theory, and K-theory.