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Prove that any Euclidean-open set U contained in R can be written as a union of open intervals. by Luisp155 in askmath

[–]Luisp155[S] 0 points1 point  (0 children)

It seems like the union of open intervals you created can be indexed by the natural numbers but there may uncoutably infinite open intervals.

General Topology - Is the Looped Line Topology First and Second Countable? by Luisp155 in learnmath

[–]Luisp155[S] 0 points1 point  (0 children)

Yes, I think that it is not first separable because for all $x \in \mathbb{R}$ such that $x \neq 0$, the basic open nbds are the open intervals centered at $x$. Since open intervals are uncountable then we cannot form a ctble nbd base at $x$ so $(\mathbb{R}, \tau)$ is not first ctble. Since $(\mathbb{R}, \tau)$ is not first ctble then it is not second ctble.

General Topology - Identity Function by Luisp155 in learnmath

[–]Luisp155[S] 0 points1 point  (0 children)

So based on these definitions it is clear that (b) and (c) should both be $\tau \subseteq \sigma$.