why $\{x\}\times Y\simeq Y$ and $X\times \{y\}\simeq X$?

Question

Fix $y\in Y$ and consider for every $x\in X$ the set, $$U_x=(\{x\}\times Y)\cup (X\times \{y\}).$$ Then every $U_x$ is connected for it is union of connected sets ($\{x\}\times Y\simeq Y$ and $X\times \{y\}\simeq X$) with non-empty intersection ($(\{x\}\times Y)\cap (X\times \{y\})=(x, y)$).

Now My QUESTION is that why $\{x\}\times Y\simeq Y$ and $X\times \{y\}\simeq X$?

My attempts : here $\{x\} \neq 0$ ..as $\{x\}\times Y\simeq \{x\}\times Y$

Im confused ...

Pliz help me

Answer 1

The space $Y$ is homeomorphic to $\{x\}\times Y$ because there is the homeomorphism$$\begin{array}{ccc}Y&\longrightarrow&\{x\}\times Y\\y&\mapsto&(x,y).\end{array}$$It is a homeomorphism because it is continuous and its inverse, which is$$\begin{array}{ccc}\{x\}\times Y&\longrightarrow&Y\\(x,y)&\mapsto&y\end{array}$$is continuous too.