A basic problem of topology is determining whether two spaces are homeomorphic. If so, then a continuous mapping between them with continuous inverse exists. Otherwise, if one can find a topological property that holds for one but not the other, then they cannot be homeomorphic:
- $\left(0,1\right)$ and $\left[0,1\right]$ are not homeomorphic because the latter is compact and the former is not.
- $\mathbb R$ and the long line are not homeomorphic because the former has a countable basis and the latter does not.
- $\mathbb R$ and $\mathbb R^2$ are not homeomorphic because deleting a point from the latter leaves a connected space while doing so from the former does not.
However, distinguishing between $\mathbb R^2$ and $\mathbb R^3$ is more involved: deleting a point from the latter leaves a simply connected space while doing so from the former does not.
To distinguish between more spaces, a more general tool, their fundamental groups, can be used: two spaces are homeomorphic if their fundamental groups are isomorphic. For example, if a space is simply connected, then its fundamental group is trivial.
These writings build the tools necessary to talk about the fundamental group.
Claim: If $h,h':X\to Y$; $k,k':Y\to Z$; $h\simeq h'$; and $k\simeq k'$, then $k\circ h\simeq k'\circ h'$.
Proof: $(x,t)\mapsto G(F(x,t),t)$ is a homotopy.
$$\tag*{$\blacksquare$}$$
Claim: $[X,I]$ is a singleton.
Proof: $(x,t)\mapsto(1-t)f(x)+tg(x)$ is a homotopy.
$$\tag*{$\blacksquare$}$$
Claim: If $Y$ is path-connected, then $[I,Y]$ is a singleton.
Proof: If $h$ is a path from $f(1)$ to $g(0)$, then $f,g\in[I,Y]$ are closed subpaths of and thus homotopic to $f*h*g$.
$$\tag*{$\blacksquare$}$$
Claim: $I$ and $\mathbb R$ are contractible.
Proof: $(x,t)\mapsto (1-t)x$ is a homotopy, i.e., $\text{id}_I$ and $\text{id}_\mathbb R$ are nulhomotopic to $0$.
$$\tag*{$\blacksquare$}$$
Claim: If $X$ is contractible, then $X$ is path-connected.
Proof: $F(x,\cdot)$ is a path from $x$ to $x_0$, where $F$ is a homotopy between $\text{id}_X$ and $x_0$.
$$\tag*{$\blacksquare$}$$
Claim: If $Y$ is contractible, then $[X,Y]$ is a singleton.
Proof: $\text{id}_Y\circ f=f$ is homotopic to $y_0\circ f=y_0$ since $\text{id}_Y$ is homotopic to $y_0$.
$$\tag*{$\blacksquare$}$$
Claim: If $X$ is contractible and $Y$ is path-connected, then $[X,Y]$ is a singleton.
Proof: $f\circ\text{id}_X=f$ is homotopic to $f\circ x_0=f(x_0)$ since $\text{id}_X$ is homotopic to $x_0$ and all constant maps are nulhomotopic since $Y$ is path-connected.
$$\tag*{$\blacksquare$}$$
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