Why don't all path-connected space deformation retract onto a point?

Question

I'm reading Hatcher's book Algebraic Topology and on page 12 he writes:

... It is less trivial to show that there are path-connected spaces that do not deformation retract onto a point...

The definition of deformation retract is given as:

A deformation retraction of a space $X$ onto a subspace $A$ is a family of maps $f_t :X→X$, $t \in I$, such that $f_0 = 1 $ (the identity map), $f_1(X) = A$, and $f_t |A = 1$ $\forall t$. The family $f_t$ should be continuous in the sense that the associated map $X×I→X, (x,t) \rightarrow f_t(x)$ is continuous.

Now my question is: how is it possible for a space $X$ to be path-connected but not retractable to a single point? Can one not just pick $x \in X$ and then retract any $y \in X$ along the path between $x$ and $y$?

Answer 1

Suppose $X$ is path-connected, then for any $y \in X$, and for any $x$, there is a path $\phi_x : I \to X$ such that $\phi_x(0)=x$ and $\phi_x(1) = y$.

With those maps, you can form a map $\phi : X \times I \to X$, with $\phi(x,t) = \phi_x(t)$. This map has all the properties you would want of a deformatino retract from $X$ onto $y$, except that it has no reason at all to be continuous.

An example of a non retractable path connected space is the circle.

Let's say you pick $y = (1,0)$, and decide to connect any $x$ to $y$ in a clockwise motion, via the top of the circle. Then the map you $\phi$ you will obtain will not be continuous at $(y,t)$ for any $t \in ]0;1[$, since the neighboor points below $y$ will travel all around the circle, while $y$ stays where he is.

And as Hatcher says, it is not trivial to show that the circle is not retractable onto a point.

Answer 2

It's probably worth pointing out that you should be careful with the term retraction versus deformation retraction. Every space $X$ retracts onto a point, via the map $r \colon X \rightarrow \ast$ that sends everything to a single point, but as Hatcher says, not every space deformation retracts to a point.

What Hatcher's definition of deformation retract is really saying is that a deformation retract is a retract $r$ along with a homotopy $r \simeq 1$, where $1$ is the identity. So if a space deformation retracts to another space, those two spaces have the same homotopy type, whereas spaces can retract onto spaces that that are not of their homotopy type (as you'll soon see if you keep reading the book).

In particular, if every space deformation retracted to a point, every space would be contractible, and algebraic topology would be very boring indeed!