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It is generally known that the concept of sequences does not yield a satisfactory theory of convergence in arbitrary topological spaces. Instead one considers more general objects such as filters. However, many theorems about filters also become true for sequences when one restricts the attention to first countable spaces. Here are some examples:

  • $x \in X$ belongs to the closure of $A \subseteq X$ iff there is a filter on $A$ which converges to $x$.
  • $f : X \to Y$ is continuous at $x$ iff $f(\Phi)$ converges to $f(x)$ whenever $\Phi$ converges to $x$.
  • $X$ is Hausdorff iff every filter converges to at most one point.
  • $X$ is countably compact iff every filter with countable basis has an accumulation point.

Is there any connection between sequences and filters in first countable spaces that explains their interchangeability in theorems like above?

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The connection would be more clearly seen between nets and seuqences in first-countable spaces. Recall that a net in a topological space $X$ consists of a directed set $\langle \Sigma , \leq \rangle$ and a mapping $\Sigma \to X$ (where $\sigma \mapsto x_\sigma$). We say that the net converges to $x$ (and that $x$ is the limit of the net) if for each neighbourhood $U$ of $x$ there is a $\sigma_0 \in \Sigma$ such that $x_\sigma \in U$ for all $\sigma \geq \sigma_0$.

Note that any sequence is clearly a net (with respect to the directed set $\langle \mathbb{N} , \leq \rangle$.)

The basic theorem is the following:

Given a topological space $X$, $A \subseteq X$ and $x \in X$, $x \in \overline{A}$ iff there is a net consisting of points of $A$ converging to $x$.

Now suppose $X$ is first countable. If $A \subseteq X$ and $x \in \overline{A}$, then there is a net $\langle x_\sigma \rangle_{\sigma \in \Sigma}$ of points of $A$ converging to $x$. Fix a descending countable neighbourhood base $\{ U_n : n \in \mathbb{N} \}$ for $x$, and construct an increasing sequence $\langle \sigma_i \rangle_{i \in \mathbb{N}}$ in $\Sigma$ as follows:

  • fix $\sigma_0 \in \Sigma$ such that $x_\sigma \in U_0$ for all $\sigma \geq \sigma_0$.
  • fix $\sigma_{i+1} \in \Sigma$ such that $\sigma_{i+1} \geq \sigma_i$ and $x_\sigma \in U_{i+1}$ for all $\sigma \geq \sigma_{i+1}$.

It is then easy to see that the sequence $\langle x_{\sigma_i} \rangle_{i \in \mathbb{N}}$ converges to $x$ (and consists of points of $A$).

The other theorems you have given with respect to filters have basic translations in terms of nets. The scheme developed above will show that for first-countable spaces it suffices to consider sequences.

So it all rest on the fact that first-countability allows you to "diagonalise" through nets in a "seqeuntial" fashion.

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