Is $R_K$ first countable?

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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$\mathbf{Rk} = \Bbb{R}^k$? – Stahl May 6 '13 at 17:06
R with k topology – junegirl May 6 '13 at 17:10
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
$K$-topology is interesting. – Paul May 19 '13 at 10:13

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.
@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
@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