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countability of $R_K$

Is $R_K$

  1. First countable?
  2. Second countable?

$R_K$ is $\Bbb R$ with the $K$-topology.

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1  
$\mathbf{Rk} = \Bbb{R}^k$? –  Stahl May 6 '13 at 17:06
    
R with k topology –  junegirl May 6 '13 at 17:10
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What have you tried? –  A.P. May 6 '13 at 17:18
    
till now all i know is RK is 2nd countable for the sets (a, b) and (a, b) − K where the intervals have rational end-points, constitute a countable basis –  junegirl May 6 '13 at 17:21
    
plz someone help me with this i gotta an end sem paper on this tomorrow :( –  junegirl May 6 '13 at 18:00
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1 Answer

Note that if $R_K$ is second countable, then it’s automatically first countable. Thus, if you can find a countable base for its topology, you can answer the entire question.

HINT: Start with a familiar countable base $\mathscr{B}$ for the usual topology and modify it by adding only countably many sets to get a countable base for the $K$-topology.

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so, B={collection of all open intervals (a,b) where a and b are rationals will do?} –  junegirl May 6 '13 at 17:26
    
@junegirl: That’ll do fine as a countable base for the usual topology; now what sets do you have to add to it to make it a base for the $K$-topology? –  Brian M. Scott May 6 '13 at 17:28
    
plz help me with this i gotta an end sem paper on this tomorrow :( –  junegirl May 6 '13 at 17:28
    
@junegirl: The base given in the Wikipedia article to which I linked your question isn’t countable, but it should give you exactly the idea that you need to modify $\mathscr{B}$ to get a countable base for the $K$-topology. –  Brian M. Scott May 6 '13 at 17:30
    
all open intervals(a,b) where a and b are rational and all sets of the form (a,b)-K, where K={1/n|n belong to Z_+} ? –  junegirl May 6 '13 at 17:34
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