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 Post subject: Not HausdorffPosted: Fri Jun 10, 2016 7:48 pm
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Joined: Tue Nov 10, 2015 8:25 pm
Posts: 314
Find an example of a space locally homeomorphic to $\mathbb{R}$, but not satisfying the Hausdorff condition.

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 Post subject: Re: Not HausdorffPosted: Sat Jun 11, 2016 5:42 pm
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Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Hi Nickos. Here is a possible answer.

Consider $\displaystyle{M=\mathbb{R}\cup\,\left\{(0,1)\right\}}$ and the sets

$\displaystyle{U_1=\left\{(t,0)\,,t\in\mathbb{R}\right\}\,\,,U_2=\left\{(t,0)\,,t\in\mathbb{R}-\left\{0\right\}\right\}\cup\left\{(0,1)\right\}}$

Also, define the maps

$\displaystyle{\phi_1(t,0)=t\,,(t,0)\in U_1}$

$\displaystyle{\phi_{2}(t,0)=t\,,t\in\mathbb{R}-\left\{0\right\}\,\,,\phi(0,1)=0}$.

We have that $\displaystyle{M=U_1\bigcup U_2\,\,,U_1\bigcap U_2=\mathbb{R}-\left\{0\right\}}$

and $\displaystyle{\phi_1\,,\phi_2}$ are homeomorphisms.

If $\displaystyle{p=(0,0)\in M\,\,,q=(0,1)\in M}$, then for each $\displaystyle{\epsilon>0}$ holds

$\displaystyle{(-\epsilon,\epsilon)\cap (-\epsilon,0)\cup (0,\epsilon)\cup\left\{0\right\}\neq \varnothing}$

and $\displaystyle{M}$ is not a $\displaystyle{\rm{Hausdorff}}$ - space.

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