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Suppose $(X,\tau)$ is a topological space which is locally metrizable, that is every point $x \in X$ has an open neighbourhood $U$ in which we can define a metric that gives rise to the same topology of $U$ (as a subset of $X$).

Suppose now that we have a function $f: (X,\tau) \to \mathbb{R}$ (or any metric space in place of $\mathbb{R}$). Can I reason with sequences in order to show that $f$ is continuous? That is, is sequential continuity equivalent to continuity in locally metrizable topological spaces?

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  • $\begingroup$ What happens when you try to prove it? Questions showing no effort may be closed here. $\endgroup$
    – GEdgar
    Dec 25, 2021 at 15:04
  • $\begingroup$ sorry, maybe I'm not getting the point here. Does this imply that I can reason with sequences when try to prove weak continuity of a functional f from a Banach space X whose dual is separable (hence the weak topology can be induced by a metric on norm-bounded subsets)? $\endgroup$
    – vampip
    Dec 25, 2021 at 16:47

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Let $X$ be locally metrisable.

If $A$ is sequentially closed, then if $x \in \overline{A}$, we'd have a metrisable neighbourhood $U_x$ of $x$ by assumption. It would easily follow that there is a sequence $a_n \in A\cap U_x$ so that $a_n \to x$ and sequential closedness of $A$ would imply $x \in A$ and hence $A$ is closed.

The essence of the proof is that a locally metrisable space is first countable.

So indeed sequential continuity with domain $X$ would imply ordinary continuity e.g. (a standard consequence of being a sequential space).

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  • $\begingroup$ Thank you! Yes, I see that it was indeed trivial. My doubt was concerning (infinite dimensional) separable Hilbert spaces, where the norm-bounded subsets are metrizable for the weak topology. Then I thought that I could reason with sequences in those spaces, to prove weak continuity, that is they are sequential spaces. But I guess my error is thinking that open neighbourhoods in the weak topology can be bounded, which they cannot be. Is that correct? $\endgroup$
    – vampip
    Dec 26, 2021 at 10:56

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