Contractible topological space
From Maths
- Not to be confused with: simply-connected topological space
Contents
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space. We say [ilmath]X[/ilmath] is a contractible topological space if^{[1]}:
- [ilmath]\exists c\in X\big[(:x\mapsto c)\simeq \text{Id}_X\big][/ilmath]
- In words, there is a constant map (in this case [ilmath](:X\rightarrow X)[/ilmath] by [ilmath](:x\mapsto c)[/ilmath] for some constant [ilmath]c[/ilmath]) that is homotopic to [ilmath]\text{Id}_X:X\rightarrow X[/ilmath] by [ilmath]\text{Id}_X:x\mapsto x[/ilmath] (the identity map on [ilmath]X[/ilmath])
We can expand this definition to:
- [ilmath]\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][/ilmath]^{[Note 1]} - where [ilmath]C(X,Y)[/ilmath] is simply the set of all continuous maps from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath]
- In words: there exists a point [ilmath]c\in X[/ilmath] and a homotopy [ilmath]H:X\times I\rightarrow X[/ilmath] such that the homotopy is the identity map of [ilmath]X[/ilmath] when [ilmath]t\eq 0[/ilmath] and the constant map mapping [ilmath]X[/ilmath] onto [ilmath]\{c\} [/ilmath] when [ilmath]t\eq 1[/ilmath]
Caveats
Be aware that if [ilmath]X[/ilmath] is contractible then each point is a deformation retraction of [ilmath]X[/ilmath] 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.Statements
- [ilmath]\forall x\in X[\{x\}\text{is a } [/ilmath][ilmath]\text{deformation retraction} [/ilmath][ilmath]\text{ of }X][/ilmath]
- In words: [ilmath]X[/ilmath] deformation retracts to any [ilmath]x\in X[/ilmath]
- See: a topological space is contractible if and only if for all points in the space the space deformation retracts onto that point
- [ilmath]X[/ilmath] is homotopy equivalent to a 1-point topological space
See also
Notes
- ↑ Usually if [ilmath]f\simeq g[/ilmath] then [ilmath]f[/ilmath] is the [ilmath]t\eq 0[/ilmath] side of the homotopy. We've flipped them here ([ilmath]t\eq 0[/ilmath] corresponds to the identity side) - this doesn't matter as [ilmath]H'(x,t):\eq H(x,1-t)[/ilmath] is of course a homotopy of [ilmath]H[/ilmath] is, this is part of the proof that homotopy of maps is an equivalence relation which shows us [ilmath]f\simeq g[/ilmath] means [ilmath]g\simeq f[/ilmath], so we're okay either way.