Equiconnected space: Difference between revisions

Revision as of 16:46, 26 October 2023

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A topological space $X$ is said to be equiconnected if there is a continuous map $k:X\times [0,1]\times X\to X$ such that $k(x,t,x)=x$ for all $x$ and $k(x,0,y)=x,k(x,1,y)=y$ for all $x$ and $y$ .

Roughly, speaking, at any given time $t$ , we get a map $X\times X$ to $X$ . At time $0$ , it is projection on the first coordinate, and at time 1, it is projection on the second coordinate. For elements on the diagonal, it always remains the value at the diagonal.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
contractible space	has a contracting homotopy	equiconnected implies contractible	contractible not implies equiconnected	\|FULL LIST, MORE INFO

References

MathOverflow question: Spaces that are contractible mod diagonal: This describes the property and asks for its name

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 ==Definition==
-A [[topological space]] <math>X</math> is said to be '''equiconnected''' if there is a continuous map <math>k:X \times I \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x</math> and <math>y</math>. Here, <math>I</math> is the unit interval <math>[0,1]</math>.
+A [[topological space]] <math>X</math> is said to be '''equiconnected''' if there is a continuous map <math>k:X \times [0,1] \times X \to X</math> such that <math>k(x,t,x) = x</math> for all <math>x</math> and <math>k(x,0,y) = x, k(x,1,y) = y</math> for all <math>x</math> and <math>y</math>.
 Roughly, speaking, at any given time <math>t</math>, we get a map <math>X \times X</math> to <math>X</math>. At time <math>0</math>, it is projection on the first coordinate, and at time 1, it is projection on the second coordinate. For elements on the diagonal, it always remains the value at the diagonal.