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Two things: Suppose that $\{ \mathcal{U}_n \}_{n \in \omega}$ and $\{ \mathcal{V}_n \}_{n \in \omega}$ are sequences of open $\omega$-covers and each $\mathcal{V}_n$ is a refinement of $\mathcal{U}_ …

Not necessarily. The ordinal space $\omega_1 = [ 0 , \omega_1 )$ provides a counterexample. To see that there is no locally finite cover by relatively compact sets, note that every compact subset — …

Note that the open subsets of (what I will denote by) $\mathbb{R}_B$ are of the form $U \cup A$ where $U \subseteq \mathbb{R}$ is open in the usual topology, and $A \subseteq B$ is arbitrary. Suppo …

The following list of results will show that $Y$ is Polish. It is basic that the continuous image of a separable space is separable. Also basic is that the open image of a first-countable space is f …

Another classical example is the Arens-Fort Space. Let $X = \omega \times \omega$. For each $A \subseteq X$ and $n \in \omega$ we let $A_n = \{ m \in \omega : (n,m) \in A \}$ denote the $n$th secti …