What does mean a space be sequentially dense?

Question

I was looking at some notes of real analysis then this appeared and I couldn't find this in anywhere. Can someone clarify what is the difference between sequential density and just density?

Answer 1

A subset $A$ of $X$ is sequentially dense in $X$ if for every $x \in X$ we can find a sequence in $A$ that converges to $x$, i.e. the sequential closure of $A$ (the set of all limits of sequences from $A$) equals $X$.

Normal density only guarantees we can find a net ( a generalised sequence) from $A$ converging to $x$ for each $x$, or that the topological closure of $A$ equals $X$.

A sequentially dense subset is always dense, but the reverse only holds in sequential spaces, which includes all first countable (and hence metric) spaces.

There are some cases in general topology where this makes a difference, one of which occurs in weak topologies in topological vector spaces, but also here: let $X = \omega_1 +1$ be the successor to the first uncountable ordinal, in its order topology. Then $A = \omega_1 \subseteq X$ is dense in $X$ but not sequentially dense, as each sequence of countale ordinals can only converge to a countable ordinal, and not to $\omega_1 \in X\setminus A$.

Another example (more in analysis style): let $X$ be the set of all real valued functions in the pointwise topology (so seen as the product $\mathbb{R}^\mathbb{R}$). Then $A$ the set of all functions with countable support (so $f \in A$ iff $f^{-1}[\mathbb{R}\setminus \{0\}]$ is at most countable) is dense in $X$ but not sequentially dense ($f_n \in A$ and $f_n \to f$ pointwise implies that the support of $f$ is also at most countable).

Answer 2

A part of this sentence is not completely true: "A sequentially dense subset is always dense, but the reverse only holds in sequential spaces, which includes all first countable (and hence metric) spaces."

Indeed, in a sequential space, a dense set is not always sequentially dense. The sequential closure of the set, obtained by adding the limits of sequences in the set, is not always the whole space. However the transfinite sequential closure, obtained by repeating again and again the process of adding limits of sequences until a fixed point is reached, is the whole space.

For instance, a point may not be the limit of a sequence of points of the dense set, but a limit of limits of sequences in the set.

However in a sequential metric space (in particular first a countable metric space), a dense set is always sequentially dense, because the sequential closure is already sequentially closed: limits of limits are already limits.