Generalizing the idea of sequences

Question

Suppose $S$ is a non-empty set with no metric defined on it. Which of the following definitions could be extended so as to still have meaning on the set $S$ with no metric? Which of the definitions could not?

$\cdot$ Sequence on $S$

$\cdot$ Constant sequence on $S$

$\cdot$ Eventually constant sequence on $S$

$\cdot$ Convergent sequence on $S$

$\cdot$ Cauchy sequence on $S$

$\cdot$ Subsequence of a sequence on $S$

I'm thinking that since a sequence is an ordered collection of objects whereby its length corresponds to the number of objects it contains, then it should still have a meaning on $S$. Also, we could still define an infinite sequence in $S$ as an ordinary function $a: \mathbb{N} \to S$, so that every natural number is assigned to a point $a(n) \in S$. And if $S$ consists of a finite sequence, then we could use the same idea but with some subset of $\mathbb{N}$ (depending on the length of $S$) as the domain. This would mean that we can still define subsequence of a sequence on $S$?

My guess is that constant sequences and eventually constant sequences would work fine too. Here we can say that a sequence $(a_n)$ in $S$ is called constant provided $\forall n \in \mathbb{N}, a_n=a_0$?

For the rest, I'm not sure. Because if $S$ doesn't have a metric defined on it, then my understanding is that we can't measure distances between elements of $S$.

For convergent sequence on $S$, perhaps we could say that a sequence is called convergent in a non-empty set $S$ with no metric defined on it provided the sequence has finite length?

I know that every inner product space is a normed space, which in turn, is a metric space. Can we extend these definitions using normed spaces? But if every normed space is a metric space and $S$ has no metric defined on it, I don't think that is an option.

Any insight would be appreciated!

Answer 1

A sequence on $S$ is a map from $\mathbb N$ to $S$. The following properties do not depend on topological properties os $S$ and can be defined for arbitrary sets $S$:

• Sequence on S
• Constant sequence on S
• Eventually constant sequence on S
• Subsequence of a sequence on S

The concept of

• Convergent sequence on S

for example can be extended to $S$ if there is a topology on $S$ even if it is not metrizable.

For every set $S$ there is the trivial metric defined

$$d(x,y)=\begin{array}\\0,\;x=y\\1,\;x\ne y\end{array}$$

and so all concepts using metrics can be extended to $S$ by using this metric.

With this metric a convergent sequence or a Cauchy sequence is an eventually constant sequence.