Hello guys. I need to prove the following:(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex]X=(\mathbb{R}\times \{0\})\cup(\mathbb{R}\times\{1\})[/itex] and [itex](x,0)\sim (x,1)[/itex] when [itex]x \neq 0[/itex]. Prove that [itex]L:=X/\sim[/itex] is a topological space locally homeomorphic to [itex]\mathbb{R}[/itex], but is not Hausdorff.

In order to prove that [itex]L[/itex] is homeomorphic to [itex]\mathbb{R}[/itex], all I need to do is show a continuous function [itex]f:L\longrightarrow \mathbb{R}[/itex] such that [itex]f[/itex] is invertible and [itex]f^{-1}[/itex] is also continuous, right?

I am new at this, so I am a bit confused on the Hausdorff part. A topological space isnotHausdorff if there is a pair of distinct points [itex]x,\,y[/itex] such that there are open sets [itex]U[/itex] and [itex]V[/itex] so that [itex]x\in U[/itex] and [itex]y\in V[/itex], but [itex]U\cap V \neq \emptyset[/itex], right?

If what I stated above is true, then I need to find two open sets, one containing the point [itex](0,0)[/itex] and the other containing [itex](0,1)[/itex], such that their intersection is not empty? Will that be sufficient?

Thx for the help and sorry for my english.

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# Question on homeomorfism

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