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Apologies if this is too low-level. A related question that I asked on the Math Stack Exchange got no answers after a year, so I thought it might be better to ask this one here. The standard approach ...

As a bonus, is every compact, $T_0$, second-countable space the image of $2^\omega \times \omega$? As a further bonus, can we strengthen "image" to "quotient"? My motivation for ...

Let $S$ be the set of injective sequences in $\mathbb{R}$: $$S = \{s: \mathbb{N} \rightarrow \mathbb{R}: s(m) \neq s(n) \text{ if }m \neq n\}.$$ Consider $S$ with the topology of pointwise convergence,...