# general topology – quotient of \$ R ^ 2 \$

Consider $$X$$=$$R ^ 2$$ and leave A $$in X$$ to be A = {$$(- 1.0), ( 1.0)$$}. Now consider the space of the quotient $${X} / {A}$$.

Is it Hausdorff? Connected? Compact?

My answer: the projection of $$X$$ at $$A$$ is surjective and continuous, $$X$$ is connected and so is $${X} / {A}$$.

The projection of the open cover of $$X$$ ball compound centered on $$(0,0)$$ is an open cover of $$X / A$$ as they are both open and saturated. We can not extract finite under-coverage and because of that $${X} / {A}$$ is not compact.

Honestly, I'm not sure of the evidence about compactness and I'm stuck with the Hausdorff question. If I could show that the projection is open, I could work from there. Working with intervals has left a proof that I am not 100% sure.