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In mathematics, particularly topology, a comb space is a particular subspace of that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space. Consider with its standard topology and let K be the set . The set C defined by: considered as a subspace of equipped with the subspace topology is known as the comb space. The deleted comb space, D, is defined by: This is the comb space with the line segment deleted. The comb space and the deleted comb space have some interesting topological properties mostly related to the notion of connectedness.

The comb space, C, is path connected and contractible, but not locally contractible, locally path connected, or locally connected.
The deleted comb space, D, is connected: Let E be the comb space without . E is also path connected and the closure of E is the comb space. As E D the closure of E, where E is connected, the deleted comb space is also connected.
The deleted comb space is not path connected since there is no path from (0,1) to (0,0): Suppose there is a path from p = (0, 1) to the point (0, 0) in D. Let f : [0, 1] → D be this path. We shall prove that −1{p} is both open and closed in [0, 1] contradicting the connectedness of this set. Clearly we have f −1{p} is closed in [0, 1] by the continuity of f. To prove that f −1{p} is open, we proceed as follows: Choose a neighbourhood V (open in R2) about p that doesn’t intersect the x–axis. Suppose x is an arbitrary point in f −1{p}. Clearly, f(x) = p. Then since f −1(V) is open, there is a basis element U containing x such that f(U) is a subset of V. We assert that f(U) = {p} which will mean that U is an open subset of f −1{p} containing x. Since x was arbitrary, f −1{p} will then be open. We know that U is connected since it is a basis element for the order topology on [0, 1]. Therefore, f(U) is connected. Suppose f(U) contains a point s other than p.

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Concepts associés (3)

Rétraction

En topologie générale et surtout en topologie algébrique, une rétraction est, intuitivement, un « rétrécissement » d'un espace topologique sur l'un de ses sous-espaces. Ce sous-espace est un rétract par déformation s'il existe une fonction permettant d'effectuer ce « rétrécissement » de façon continue. Soient X un espace topologique et A un sous-espace. Une rétraction de X sur A est une application continue r de X dans A dont la restriction à A est l'application identité de A, c'est-à-dire telle que pour tout point a de A, r(a) = a ; autrement dit, c'est une rétraction de l'application d'inclusion i de A dans X : r ∘ i = Id.

Courbe sinus du topologue

vignette|La fonction sin(1 / x). En mathématiques, la courbe sinus du topologue est un exemple d'espace topologique connexe mais ni localement connexe, ni connexe par arcs. Elle s'obtient comme courbe représentative d'une fonction dont l'expression fait intervenir la fonction sinus. La courbe sinus fermée du topologue est l'adhérence de cette courbe dans le plan euclidien, et constitue un espace compact satisfaisant des propriétés analogues.

List of topologies

The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property. Discrete topology − All subsets are open. Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.