Not Hausdorff

General Topology
Post Reply
Tsakanikas Nickos
Community Team
Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Not Hausdorff

#1

Post by Tsakanikas Nickos »

Find an example of a space locally homeomorphic to $\mathbb{R}$, but not satisfying the Hausdorff condition.
Papapetros Vaggelis
Community Team
Posts: 426
Joined: Mon Nov 09, 2015 1:52 pm

Re: Not Hausdorff

#2

Post by Papapetros Vaggelis »

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.
Post Reply

Create an account or sign in to join the discussion

You need to be a member in order to post a reply

Create an account

Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute

Register

Sign in

Who is online

Users browsing this forum: No registered users and 14 guests