Countably compact, Lindel"of, sequentially compact, limit point compact and $\sigma$-compact spaces, and relatively compact subsets

Let $(X, \mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), with open covers, subcovers, finiteness and compactness as in Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, and finite, at most countable and uncountable as in Finite, countably infinite, countable, uncountable.

• $(X, \mathcal{T})$ is countably compact when every open cover of $X$ that is at most countable has a finite subcover.
• $(X, \mathcal{T})$ is Lindelöf when every open cover of $X$ has an at most countable subcover.
• $(X, \mathcal{T})$ is sequentially compact when every sequence $(x_k)$ in $X$, that is every function $x : \mathbb{N} \to X$ (Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure), has a subsequence $(x_{n_j})$ converging to a point of $X$, the index map $n : \mathbb{N} \to \mathbb{N}$ being strictly increasing (Sequences of reals: bounded, eventually, frequently, tails, subsequences, A strictly increasing index map satisfies $n_k \ge k$).
• $(X, \mathcal{T})$ is limit point compact when every infinite subset $A \subseteq X$ has a limit point in $X$, that is a point $p \in X$ every neighbourhood $N$ of which satisfies $N \cap (A \setminus \{p\}) \ne \varnothing$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space). Here infinite means not finite in the sense of Finite, countably infinite, countable, uncountable.
• $(X, \mathcal{T})$ is $\sigma$-compact when there is an at most countable family $\mathcal{K}$ of compact subsets of $X$ with $X = \bigcup \mathcal{K}$.
• A subset $A \subseteq X$ is relatively compact in $X$ when its closure $\overline{A}$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space) is a compact subset of $X$.

A subset $A \subseteq X$ is called countably compact, Lindelöf, sequentially compact, limit point compact or $\sigma$-compact when the subspace $(A, \mathcal{T}_A)$ is (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace), exactly as for compactness. Relative compactness is the exception and is deliberately not of that form: it is a statement about $A$ inside $X$, since $\overline{A}$ is computed in $X$, and a set may be relatively compact in one space and not in another that contains it. Every other notion on this list is intrinsic to the subspace.

The countable covers may be listed. A nonempty at most countable family $\mathcal{U}$ admits a surjection $\mathbb{N} \to \mathcal{U}$ (A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$), so countable compactness says: for every sequence $(U_n)_{n \in \mathbb{N}}$ of open sets with $X = \bigcup_{n \in \mathbb{N}} U_n$ there are finitely many indices whose sets already cover $X$. That surjection is produced from the countability assumption alone and no choice principle is involved; the empty family covers only the empty space, which is compact anyway.

Indexing starts at $0$. A sequence here is a function on $\mathbb{N}$ and $\mathbb{N}$ contains $0$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences), so a subsequence is $(x_{n_j})_{j \in \mathbb{N}}$ with $n_0 < n_1 < \cdots$ and $n_j \ge j$ (A strictly increasing index map satisfies $n_k \ge k$). An index range taken from a text that starts at $1$ must be shifted before it is used here.

Agreement with the metric definitions. Let $(X,d)$ be a metric space carrying its metric topology $\mathcal{T}_d$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not). Then the three notions that Countably compact, sequentially compact and limit point compact metric spaces defines metrically are the three defined above, read in $(X, \mathcal{T}_d)$:

• Countably compact. The open sets used there are the members of $\mathcal{T}_d$, so the at most countable open covers are the same families and the condition is the same condition, exactly as for compactness itself (For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide).
• Sequentially compact. Convergence of a sequence in the metric sense and in the sense of Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure agree on a metric topology, because the balls around a point are a neighbourhood base at it (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not); the subsequences quantified over are the same.
• Limit point compact. A point $p$ is a limit point of $A$ in the metric sense when every ball around $p$ meets $A \setminus \{p\}$, and in the sense above when every neighbourhood does; the same neighbourhood base makes the two conditions one (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, Interior, closure, boundary, exterior, derived set and isolated point in a topological space).

So no statement below about a metrizable space introduces a second notion, and every theorem of the metric development about these three properties may be quoted here once a metric inducing the topology is named. Lindelöfness, $\sigma$-compactness and relative compactness have no metric counterpart in this library and are defined here for the first time.

None of the conditions listed above is compactness by definition. Countable compactness restricts the covers tested; Lindelöfness weakens the conclusion from finite to at most countable; sequential compactness speaks about sequences instead of covers; limit point compactness speaks about subsets; $\sigma$-compactness asks only that the space be assembled from at most countably many compact pieces; relative compactness is a condition on a subset of an ambient space. Which implications hold between them, and which need a choice principle, is Compact implies countably compact, Lindel"of and limit point compact; countably compact together with Lindel"of implies compact; and, at the cost of countable or dependent choice, sequentially compact implies countably compact, countably compact implies limit point compact, and the converse holds when every singleton is closed; that some of them fail to be equivalent is witnessed by the false statements at the end of this page.

Why $\sigma$-compactness is not a compactness property at all. The real line is $\sigma$-compact, being the union of the compact intervals $[-n, n]$, and it is not compact; the definition is useful precisely because it names a class of spaces built out of compact pieces without being compact. The same remark explains why a $\sigma$-compact space need not be countably compact.

Limit point compactness is sometimes called the Bolzano-Weierstrass property, and countably compact is occasionally used for what is called limit point compact here. This library uses the four names above with the meanings given, and writes the condition out whenever the risk of confusion is real.