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Math Help - problem in topology

  1. #1
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    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
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Are you saying that your space is \mathbb{R} with the topology \left\{\mathbb{R},\varnothing\right\}\cup\left\{(-a,a):a>0\right\}?

    Quote Originally Posted by nice rose View Post
    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 \mathbb{Q} is countable. Now, let x\in X and let U be any neighborhood of it. Clearly, we may find some (-a,a) such that x\in (-a,a)\subseteq U. And so, (-a,a) contains...


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


    Hint:
    Spoiler:

    Let x\in X, what about \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 \left\{(-q,q):q\in\mathbb{Q}\right\}

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


    is (X,T) compact space ?
    Hint:

    Spoiler:
    X=\bigcup_{\alpha>0}(-\alpha,\alpha) suppose this had a finite subcover (-\alpha_1,\alpha_1),\cdots,(-\alpha_n,\alpha_n),\text{ }\alpha_1\leqslant\cdots\leqslant\alpha_n then is \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.
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  3. #3
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    Thank you so much
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