Equiconnected space: Difference between revisions

Latest revision as of 22:19, 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 nonempty 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 $x,y\in X$ .

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

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nonempty topologically convex space	nonempty space that is homeomorphic to a convex subset of Euclidean space	topologically convex implies equiconnected		\|FULL LIST, MORE INFO

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

@@ Line 3: / Line 3: @@
 ==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>.
+A nonempty [[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,y \in X</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.
 ==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::nonempty topologically convex space]] || nonempty space that is homeomorphic to a [[convex subset of Euclidean space]] || [[topologically convex implies equiconnected]] || || {{intermediate notions short|equiconnected space|topologically convex space}}
+|}
 ===Weaker properties===
@@ Line 16: / Line 24: @@
 | [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[equiconnected implies contractible]] || [[contractible not implies equiconnected]] || {{intermediate notions short|contractible space|equiconnected space}}
 |}
 ==References==
 * {{mathoverflow|number = 457103|title = Spaces that are contractible mod diagonal}}: This describes the property and asks for its name