It turns out that we are not just interested in spaces of continuous maps, but we are concerned with path-connected components of these spaces, or like things that are continuous but also can deform the thing.

Consider the maps $X \xrightarrow{f_1} Y$ and $X \xrightarrow{f_0} Y$. We say $f_0 \sim f_1$ are homotopic if there is some $F : X \times I \rightarrow Y$ such that $F(x,0) = f_0$, $F(x,1) = f_1$.

The classic example to distinguish path connectedness and connectedness is the union of $\sin(1/x)$ and the $y$-axis, which is supposedly connected but not path connected.

We say that $X$ and $Y$ are homotopy equivalent if $g \circ f \sim \text{id}_X$ (homotopic to the identity on $X$), and $f \circ g \sim \text{id}_Y$.

We define a retraction $r : X \twoheadrightarrow A\subseteq X$ so that $r|_A = \text{id}_A$. A retraction is deformational if $i \circ r \sim \text{id}_X$. Strict if the homotopy is identical on $A$, like for example, $A$ doesn’t rotate or something during the retraction and stuff. In other words, strict means that $A$ doesn’t move during the retraction. Apparently there are some pathological examples, for which one can refer to the book.

Proof.

$i \circ r \sim \text{id}_X$, $r \circ i = \text{id}_A$.

Now $X$ is called contractible if $x_0 \in X$ is a deformtaional retraction of $X$.

There is a classification of homotopies or something:

We start by showing that 1 implies 3. If $X \sim Y$, then there exist maps $f : X \rightarrow Y$ and $g : Y \rightarrow X$ such that the compositions are identity. Then we get maps:

Now we can do the direction that 3 implies 1. For this direction, he started by taking the image of the identity of $\pi(X,X)$ in $\pi(X,Y)$ under the map $\beta_X$ which was defined previously as $f_*$ I think (so that $\beta_X(id) = f$). Awesome proof using the square $\pi(X,X)$, $\pi(X,Y)$, $\pi(Y,X)$, $\pi(Y,Y)$ basically by tracing where the elements $\text{id}_X$ and $\text{id}_Y$ go to throughout the diagram. We then do a similar square for the other way.