= Topology
{wiki}
Topology studies <topological spaces> through <open sets> and <continuous maps>. Its central equivalence is a <homeomorphism>: a <bijection> that preserves <continuity> in both directions. This lets us distinguish spaces by <connectedness> or <compactness> even when numerical <metric> values are irrelevant. The <open interval> $(0,1)$ and the <real line> have the same <topology> in this sense, although only the former is a <bounded set> with its usual <metric>.
\Image[/topological-continuity-examples.png]
{title=Homeomorphisms, inverse images, and the effect of removing a point}
The upper panels show an explicit <homeomorphism> and the <inverse image> test for a <continuous map>. The lower panels exhibit an obstruction to a <homeomorphism>: removing any point from a <circle> leaves a <connected space>, whereas removing an interior point from a <closed interval> gives two <connected components>.
<Homotopy> is a different notion: it varies a <continuous map> continuously with a parameter and may collapse points. For example, $H(x,t)=(1-t)x$ contracts $[0,1]$ to zero. Thus this <closed interval> and a <singleton> have the same <homotopy type>, but there is no <homeomorphism> between them because there is no <bijection>. <Homotopy equivalence> need not preserve <compactness>. https://ocw.mit.edu/courses/18-905-algebraic-topology-i-fall-2016/2dad3dd8354992d877fbeca4049817ee_MIT18_905F16_lecture_notes.pdf[MIT Algebraic Topology I, Lecture 5]{ref}
The abstract viewpoint already appears explicitly in Tietze's 1923 paper: he describes <topology> as the study of properties invariant under invertible <continuous maps> with <continuous> inverses, and discusses axioms for <neighbourhoods>, citing Hausdorff's 1914 book, page 213. https://gdz.sub.uni-goettingen.de/fulltext/PPN235181684_0088/00000294?_format=xml[Tietze, Beiträge zur allgemeinen Topologie I, page 290]{ref}
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