
Path connected
If X and Y are path connected, show that XxY is also path connected.
For a fixed $\displaystyle x_0\in X$ and $\displaystyle y_0\in Y$
Let A be the set of all the points that join $\displaystyle x_0\times y_0$ by a path in $\displaystyle U\times V$.
Since X and Y are both path connected, there exist $\displaystyle \epsilon >0$ such that $\displaystyle B(x,\epsilon) \ \text{and} \ B(y,\epsilon)$, where x is in $\displaystyle \pi_1(A)$ and y is in $\displaystyle \pi_2(A)$, contained in open sets $\displaystyle U\subset X$ and $\displaystyle V\subset Y$, respectively.
Let $\displaystyle x_1\in B(x,\epsilon)$ then a path can be connected from $\displaystyle x_1$ to x to $\displaystyle x_0$.
Similarly for y
How can now show XxY is path connected?
How can I translate this to the product topology?
Can I say:
$\displaystyle x_1\times y_1\in B(x\times y,\epsilon)\subset A\subset U\times V$. Therefore the exist a path from $\displaystyle x_1\times y_1$ to $\displaystyle x\times y$ to $\displaystyle x_0\times y_0$????

Re: Path connected
Suppose X and Y are path connected. By definition, for any $\displaystyle x_1,x_2\in X$ there is a continuous function $\displaystyle f:[0,1]\rightarrow X$ with $\displaystyle f(0)=x_1$ and $\displaystyle f(1)=x_2$
Now let $\displaystyle (x_0,y_0)$ and $\displaystyle (x_1,y_1)$ be two points in $\displaystyle X\times Y $and let $\displaystyle f$ be the path in $\displaystyle X$ from $\displaystyle x_0$ to $\displaystyle x_1$ and $\displaystyle g$ be the path in $\displaystyle Y$ from $\displaystyle y_0$ to $\displaystyle y_1$. Define $\displaystyle h:[0,1]\rightarrow X\times Y$ by $\displaystyle h(s) = (f(s),g(s))$. Then $\displaystyle h(0)=(x_0,y_0)$ and $\displaystyle h(1)=(x_1,y_1)$.
You should know a theorem that tells you $\displaystyle h$ is continuous since it maps into a product space and each coordinate is given by a continuous function(Thm 19.6 in Munkres)