Contractible topological space

Not to be confused with: simply-connected topological space

Definition

Let $(X,\mathcal{ J })$ be a topological space. We say $X$ is a contractible topological space if:

• $\exists c\in X\big[(:x\mapsto c)\simeq \text{Id}_X\big]$
  • In words, there is a constant map (in this case $(:X\rightarrow X)$ by $(:x\mapsto c)$ for some constant $c$) that is homotopic to $\text{Id}_X:X\rightarrow X$ by $\text{Id}_X:x\mapsto x$ (the identity map on $X$)

We can expand this definition to:

• $\exists c\in X\exists H\in C(X\times I;\ X)\big[(\forall x\in X[H(x,0)\eq x])\wedge(\forall x\in X[H(x,1)\eq p])\big]$ - where $C(X,Y)$ is simply the set of all continuous maps from $X$ to $Y$
  • In words: there exists a point $c\in X$ and a homotopy $H:X\times I\rightarrow X$ such that the homotopy is the identity map of $X$ when $t\eq 0$ and the constant map mapping $X$ onto $\{c\}$ when $t\eq 1$

Caveats

Be aware that if $X$ is contractible then each point is a deformation retraction of $X$ certainly (this is in fact an equivalent statement, and given in equivalent definitions below)

• However each point need not be a strong deformation retract
• See: Example:A contractible space that does not strongly deformation retract to any point
  • This example is closely related to Example:A deformation retraction where there is no strong deformation retraction

Equivalent definitions

There are 2 other forms commonly seen as definitions for a contractible space, they are easily seen to be equivalent and are "intuitively" just as good for a definition, so we list them as equivalent definitions rather than equivalent statements.

TODO: Find a reference for them Alec (talk) 20:00, 24 April 2017 (UTC)

Statements

1. $\forall x\in X[\{x\}\text{is a }$$\text{deformation retraction}$$\text{ of }X]$
   • In words: $X$ deformation retracts to any $x\in X$
   • See: a topological space is contractible if and only if for all points in the space the space deformation retracts onto that point
2. $X$ is homotopy equivalent to a 1-point topological space
   • See: a topological space is contractible if and only if the space is homotopy equivalent to a 1-point space

See also

• Star-shaped topological space