Define a topology on three points which is contractable

Question

I don't know how should i define a homotopy on a set. I think {{},{a,b,c}} should work but i don't know how to write the homotopy between the identity map and a constant map. (So sorry for this basic quistion.....)

Answer 1

So if I understand well... You have a set $X=\{a,b,c\}$ only containing three points. You want to define a topology $\mathcal T$ on $X$ such that $X$ is contractile.

If that is the question, then indeed the trivial topology $\mathcal T =\{\emptyset, \{a,b,c\}\}$ is convenient. Why?

Consider the map defined by $$H(x,t)=\begin{cases} x & \text{for } &x \in \{a,b,c\} & \text{ and } 0 \le t <1/2\\ a & \text{for } &x \in \{a,b,c\} & \text{ and } 1/2 \le t \le 1 \end{cases}$$

It is sufficient to prove that $H$ is continuous as $H(\cdot,t)$ is a map between of the identity of $X$ and a point, namely $a$.

But this is trivial as the inverse image of the emptyset is the emptyset while the inverse image of $X$ is $X \times [0,1]$ which are both open sets of the product $X \times [0,1]$ endowed with the product topology.

Answer 2

Hint: In the trivial topology you mentioned, any map $X\to\left\{a,b,c\right\}$ is continuous.