Prove that $S^2 \vee S^2$ is path connected?

Question

Prove that $S^2 \vee S^2$ is path connected?

Let $a$ and $b$ be two points in a topological space $X$. A path in $X$ from $a$ to $b$ is a continuous map $f$ from [0,1] to $X$ s.t $f$(0) = $a$ and $f$(1)=$b$.

A topological space X is said to be path connected iff given any two points $a$ and $b$ , there exists a path in $X$ from a to b

I can draw but i can’t give a function.

Answer 1

Let $p$ be the joining point, which belongs to both copies of $S^2$.

Let $x_1,x_2\in S^2\vee S^2$. There exist paths $f_1, f_2$ such that $f_i(0)=x_i$ and $f_i(1)=p$. Then consider the path

$$F(t) = \begin{cases} f_1(2t) \ , \ t\in[0,.5) \\ f_2(2-2t) \ , \ t\in[0.5,1]\end{cases}$$

Note, $F(t)$ is sometimes called the path composition of $f_1, f_2^{-1}$ and might be notated $f_1*f_2^{-1}$ (the "inverse" because here I traverse $f_2$ backwards).

There was also nothing special about $S^2$ here.