Number of connected components quotient space

Question

Be $=\{(,)∈ℝ^2:=0 \text{ or } =1\}$ with topology $T$ induced by the usual of $\mathbb R^2$. Let be $$ equivalent relation in $$ with equivalent classes are $$[(0,0)]=[(0,1)]=\{(0,0),(0,1)\},\quad [(,)]=\{(,)\} \text{ if } ≠0.$$

Let be $(/,/)$ the quotient space and $:→/$ the projection. I want to know the number of connected components of $/∖(0,0)$.

I think they should be $3$, $1$, $4$ or $2$ but I can't see it.

Any help is appreciated!

Answer 1

The four connected components of $Y = X/R\setminus\{[0,0]\}$ are $$ \pi( \mathbb R_{<0} \times \{0\} ),\ \pi( \mathbb R_{>0} \times \{0\} ),\ \pi( \mathbb R_{<0} \times \{1\} ),\ \pi( \mathbb R_{>0} \times \{1\} ). $$ Do you see how these are all open and closed in $Y$, connected and their union is disjoint and all of $Y$?

Here $\mathbb R_{<0}$ and $\mathbb R_{>0}$ denote the sets of negative and positive real numbers, respectively.