Continuous Functions
A function f : X→ Y between topological spaces is called continuous if for all x ∈ X and all neighbourhoods N of f(x) there is a neighbourhood M of x such that f(M) ⊆ N. This relates easily to the usual definition in analysis. Equivalently, 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, Top, the category of topological spaces with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in mathematics. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated and generated entire areas of research, such as homotopy theory, homology theory, and K-theory, to name just a few.
Read more about this topic: Topological Space
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