# Generalizing the idea of sequences

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!

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.