Editing Topological space (section)

== Continuous functions ==
{{main|Continuous function}}

A [[Function (mathematics)|function]] <math>f : X \to Y</math> between topological spaces is called '''[[Continuity (topology)|continuous]]''' if for every <math> x \in X</math> and every neighbourhood <math>N</math> of <math>f(x)</math> there is a neighbourhood <math>M</math> of <math>x</math> such that <math>f(M) \subseteq N.</math> This relates easily to the usual definition in analysis. Equivalently, <math>f</math> is continuous if the [[inverse image]] of every open set is open.{{sfn|Armstrong|1983|loc=theorem 2.6}}  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 function|inverse]] is also continuous.  Two spaces are called {{em|homeomorphic}} if there exists a homeomorphism between them.  From the standpoint of topology, homeomorphic spaces are essentially identical.<ref>{{Cite book|isbn = 978-93-325-4953-1|last = Munkres|first = James R|title = Topology|date = 2015|pages = 317–319| publisher=Pearson }}</ref>

In [[category theory]], one of the fundamental [[Category (mathematics)|categories]] is '''Top''', which denotes the [[category of topological spaces]] whose [[Object (category theory)|objects]] are topological spaces and whose [[morphism]]s are continuous functions.  The attempt to classify the objects of this category ([[up to]] [[homeomorphism]]) by [[Invariant (mathematics)|invariant]]s has motivated areas of research, such as [[Homotopy|homotopy theory]], [[Homology (mathematics)|homology theory]], and [[K-theory]].