I came to this question when I was solving a problem in Willard's General Topology (p. 36).
I am new to topology and I want to prove that the looped line topology is Tychonoff space. My idea was to prove that it is a $T_4$ space. I proved that this space is hausdorff space but I'm having trouble with proving that it is a normal space. I have no idea how to do this. Any help is appreciated.
For clarity, the looped line topology is defined as follows.
- At each point $x$ of the real line other than the origin, the basic neighbourhoods of $x$ are the usual open intervals centred at $x$.
- Basic neighbourhoods of the origin are the sets of the form $$ (−\epsilon,\epsilon)\cup(−\infty,−n)\cup(n,\infty), $$ for all possible choices of $\epsilon>0$ and $n\in\Bbb N$.
Depending on what I'm allowed to assume, I'd approach this by showing that that this space is metrizable. (In fact, though I won't show it, it is a copy of a "figure eight" in the plane.) I'll give a proof of metrizability, particularly since this is not the proof that would likely be expected if this was assigned as homework.
To show this space is metrizable, and since you already know it is Hausdorff, I'll show that it is regular and second-countable.
Second-countability is easy: a countable base would include each $(p,q)$ and $(-q,-p)$ for positive rationals $p<q$, and $(-\infty,-n)\cup(-p,p)\cup(n,\infty)$ for positive rational $p$ and positive integer $n$.
For regularity, we should show that for each point $x$ and each basic open neighborhood $U$, there is another neighborhood $V$ of $x$ with $cl(V)\subseteq U$. For $x>0$, let $(p,q)$ be positive rationals with $p<x<q$. Then $\left(\frac{p+x}{2},\frac{x+q}{2}\right)$ is a neighborhood of $x$ and $cl\left(\frac{p+x}{2},\frac{x+q}{2}\right)=\left[\frac{p+x}{2},\frac{x+q}{2}\right]\subseteq(p,q)$. The case for $x<0$ is similar. Finally, given the neighborhood $(-\infty,-n)\cup(-p,p)\cup(n,\infty)$ for $0$, we have $(-\infty,-n-1)\cup(-p/2,p/2)\cup(n+1,\infty)$ with $$cl\Big((-\infty,-n-1)\cup(-p/2,p/2)\cup(n+1,\infty)\Big)$$ $$=(-\infty,-n-1]\cup[-p/2,p/2]\cup[n+1,\infty)$$ $$\subseteq(-\infty,-n)\cup(-p,p)\cup(n,\infty)$$