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)
- .... 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)
- .... 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)
- 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)
- 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).
- 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 ?