1. ## problem in topology

X = R
T = {{},R} U {(-a,a): a > 0}

is (X,T) separable space ?
is (X,T) First Countable space ?
is (X,T) Second Countable space ?
is (X,T) compact space ?
is (X,T) locally compact space ?

thanks

2. Are you saying that your space is $\displaystyle \mathbb{R}$ with the topology $\displaystyle \left\{\mathbb{R},\varnothing\right\}\cup\left\{(-a,a):a>0\right\}$?

Originally Posted by nice rose
X = R
T = {{},R} U {(-a,a): a > 0}

is (X,T) separable space ?
What do you think?

Hint:

Spoiler:
I hope you know that $\displaystyle \mathbb{Q}$ is countable. Now, let $\displaystyle x\in X$ and let $\displaystyle U$ be any neighborhood of it. Clearly, we may find some $\displaystyle (-a,a)$ such that $\displaystyle x\in (-a,a)\subseteq U$. And so, $\displaystyle (-a,a)$ contains...

is (X,T) First Countable space ?
So, does every point have a countable neighborhood base?

Hint:
Spoiler:

Let $\displaystyle x\in X$, what about $\displaystyle \left\{\left(-x-\tfrac{1}{n},x+\tfrac{1}{n}\right):n\in\mathbb{N}\ right\}$

is (X,T) Second Countable space ?
I mean, the idea is pretty straightforward. Think about "countable endpoints"

Hint:

Spoiler:

What about $\displaystyle \left\{(-q,q):q\in\mathbb{Q}\right\}$

This is clearly countable and given any point $\displaystyle x\in X$ and any neighborhood $\displaystyle x\in (-a,a)$. Now, since $\displaystyle a>x$ there exists some $\displaystyle x<q<a$ and so...

is (X,T) compact space ?
Hint:

Spoiler:
$\displaystyle X=\bigcup_{\alpha>0}(-\alpha,\alpha)$ suppose this had a finite subcover $\displaystyle (-\alpha_1,\alpha_1),\cdots,(-\alpha_n,\alpha_n),\text{ }\alpha_1\leqslant\cdots\leqslant\alpha_n$ then is $\displaystyle \alpha_n+1\in(-\alpha_1,\alpha_1)\cup\cdots\cup(-\alpha_n,\alpha_n)$?

is (X,T) locally compact space ?

thanks
No hint on this one, I want you to try it for yourself.

3. Thank you so much