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
Are you saying that your space is $\displaystyle \mathbb{R}$ with the topology $\displaystyle \left\{\mathbb{R},\varnothing\right\}\cup\left\{(-a,a):a>0\right\}$?
What do you think?
Hint:
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So, does every point have a countable neighborhood base?is (X,T) First Countable space ?
Hint:
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I mean, the idea is pretty straightforward. Think about "countable endpoints"is (X,T) Second Countable space ?
Hint:
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Hint:is (X,T) compact space ?
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No hint on this one, I want you to try it for yourself.is (X,T) locally compact space ?
thanks