"Countable" and "Sequence" in Toplology

Question

I'm having difficulty with the meanings of these terms in several references. "Countable" may mean either a set with cardinality = N (i.e. countably infinite) , or with cardinality $\le$ N (i.e. countably infinite or finite). "Sequences" may mean specifically an infinite sequence (with cardinality = N), or may allow infinite or finite (with cardinality $\le$ N)

1. .... Specifically, a space X is said to be first-countable if each point has a countable neighborhood basis (local base). That is, for each point x in X there exists a sequence $N_1, N_2, … $ of neighborhoods of x such that ....(Wiki)
2. .... More explicitly, this means that a topological space T is second countable if there exists some countable collection $\mathcal{U} = \{U_i\}_{i=1}^\infty $ of open subsets of T... (Wiki)
3. a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.... (Wiki)
4. In mathematics a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence $\{x_{n}\}_{n=1}^{\infty } $of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.... (Wiki - in this case it seems that the sequence is intended to be infinite).
5. In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. ....Some authors use countable set to mean countably infinite alone (Wiki)

My confusion continues into, for example, the proof that a second countable metric space is Lindelöf, e.g. here: https://proofwiki.org/wiki/Sequentially_Compact_Metric_Space_is_Lindel%C3%B6f As far as I can see, the proof seems to establish that every cover has a "countable", possibly finite, subcover: should it also prove that the subcover is not finite ?

Answer 1

I don't follow what the problem is. In general, and here, countable means countably infinite or finite. Sequence (at least in this context) means "sequence indexed by the natural numbers". Maybe you need to note that the terms in a sequence need not be distinct; an "infinite sequence" may actually attain only finitely many values.

So for example say $X=\{0\}$, with the only possible topology. Then $X$ is first countable. The required sequence of neighborhoods of $0$ is $N_1,N_2,\dots$, where $N_j=X$ for all $j$.

Similarly a space with only finitely many points is separable. It's clear that such a space contains a countable dense subset. The sequence in the "that is" part of (4) would be any ("infinite") sequence that contains every element of $X$ infinitely often.