Show that every contractible space is connected.

Question

So I'm trying to show every contractible topological space is connected. At the moment I've established that $X$ is contractible iff there is a homotopy equivalence $f:\{x_0\}\to X$ for some $x_0\in X$, but I'm having trouble showing that if $X$ and $Y$ are homotopy equivalent and $X$ is connected, then so is $Y$ (from which the result would follow, as $\{x_0\}$ is clearly connected).

Answer 1

Let $H: X \times I$ be a contracting homotopy to $p \in X$, so that

• $H$ is continuous.
• $H(x,0)=x$ for all $x \in X$.
• $H(x,1)=p$ for all $x \in X$.
• $H(p,t)=p$ for all $t$.

Then for each $x \in X$ we have a continuous path $p_x: [0,1] \to X$ such that $p_x(0)=x, p_x(1)= p$, namely $p_x(t)=H(x,t)$ where we restrict $H$ to $\{x\} \times I$, reallly, so $p_x$ is continuous as $H$ is.

Then clearly $X$ is path-connected: If $x,y \in X$ define $p: I \to X$ by

$$p(t)= \begin{cases} p_x(2t)& 0 \le t \le \frac12\\ p_y(2-2t) & \frac12 \le t \le 1\\ \end{cases}$$

which is continuous by the pasting lemma for two closed sets (note that $t=\frac12$ is consistent as $p_x(1)= p = p_y(1)$, so they connect); we go from $x$ to $p$ via $p_x$ and then beckwards from $p$ to $y$ via $p_y$.

And a path-connected space is connected. (Follows from $I$ being connected and continuous paths preserving connectedness, so $p[I]$ also is connected.)

Answer 2

I'll write an outline to what I think works.

Let $H(x,t):X\times [0,1]\to X$ be the homotopy contracting the space to a point $x_0$. I will just show that for all $x\in X$ there exists a path $\gamma_x:[0,1]\to X$ such that $\gamma_x(0)=0$ and $\gamma_x(1)=x_0$. I leave it to construct a path from $x_0$ to any $y\in X$ and that create a path $\gamma_{x,y}:[0,1]\to X$ such that $\gamma_{x,y}(0)=x$ and $\gamma_{x,y}(1)=y$.

Define $\gamma_x(t):=H(x,t)$ and recall that $H(\cdot,0)=Id_X$, hence $\gamma_x(0)=x$. Since $H$ is continuous, the projection $\pi_{I}(x,t)=t$ is continuous, and $\gamma_x(t)=H(x,t)$ is continuous.