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Math Help - countable first countable

  1. #1
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    countable first countable

    True/False:
    1. Is every countable first countable T_3-space metrizable?
    2. Is every separable first countable T_3 space metrizable?

    Attempt
    1. countable + first countable \Rightarrow second countable. second countable + T_3 \Leftrightarrow T_4. So by Urysohn Theorem, since this is second countable + T_4, this is metrizable. But this argument doesn't work b/c T_3 \not \Rightarrow T_4 in general. So is this false? What is the counterexample?
    2. separable + first countable \Rightarrow second countable. Now, the argument is the same as above.
    Thanks.
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  2. #2
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    Quote Originally Posted by xianghu21 View Post
    True/False:
    1. Is every countable first countable T_3-space metrizable?
    2. Is every separable first countable T_3 space metrizable?

    Attempt
    1. countable + first countable \Rightarrow second countable. second countable + T_3 \Leftrightarrow T_4. So by Urysohn Theorem, since this is second countable + T_4, this is metrizable. But this argument doesn't work b/c T_3 \not \Rightarrow T_4 in general. So is this false? What is the counterexample?
    2. separable + first countable \Rightarrow second countable. Now, the argument is the same as above.
    Thanks.
    The Uryshon Metrization Theorem: Every second countable regular Hausdorff space is metrizable.

    For (1), if the topological space X is countable, then a countable basis can cover X. Thus, X is second countable (If X is second countable, then it is automatically first countable). By Urysohn's metrization theorem, every second countable regular Hausdorff space is metrizable (I denoted T_3 as a regular Hausdorff space. Otherwise, you can also use the fact that every regular Lindelof space is normal. This issue is discussed here).
    Thus, (1) is correct.

    For (2), the counterexample can be a lower limit topology on R, denoted as R_l. R_l has a countable local basis at each point x \in R_l, where \{[x, x+1/n)\}_{n=1}^{\infty} is a countable local basis at x. It is separable since the rational numbers are dense in R_l.
    R_l has no countable basis. Let B be a basis for R_l. For each x in R_l, we choose a basis element B_x of B such that x \in B_x \subset [x, x+1). If x \neq y, then B_x \neq B_y, since x=glb \text{ } B_x and y=glb \text{ } B_y. Thus, B must be uncountable. It follows that R_l is not second countable.
    Thus, (2) is not correct.

    Some additional remarks: The regularity hypothesis in the Urysohn metrization theorem is a necessary one, but the second countable condition is not. You can use other conditions to check metrizability (link).
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