The four live convention forks of general topology and which side this library takes on each

General topology is a subject whose textbooks disagree with one another on vocabulary far more than on content. Four of those disagreements are live inside this page, in the sense that a reader arriving with the other convention would misread a statement here rather than merely find it unfamiliar. Each is settled below, once, and the settlement is used without further comment everywhere on these two pages. Where this library's choice is the less common one it is said so.

1. A neighbourhood need not be open. A set $N$ is a neighbourhood of $x$ when some open $U$ satisfies $x \in U \subseteq N$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open). The competing convention, used by Munkres among many others, reserves the word for open sets containing $x$. A condition quantified over every neighbourhood is equivalent to its restriction to open neighbourhoods when the condition is preserved on enlarging the set, as eventual-membership and the standard local tests are; this is not true for an arbitrary predicate. The wider notion is chosen because it makes the neighbourhoods of a point a filter, and because a neighbourhood base is then allowed to consist of sets that are not open. This library writes open neighbourhood in full whenever openness is being used.

2. The empty intersection is the whole set, and a subbasis need not cover. In A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis the finite intersections of a subbasis $\mathcal{S}$ include the intersection of no members, which is $X$; consequently $X$ is always basic, the criterion (B1) is automatic, and no covering hypothesis $\bigcup \mathcal{S} = X$ is imposed on a subbasis (Basis and subbasis for a topology, and the topology generated by a family of sets). The competing convention admits only nonempty finite intersections and adds the covering hypothesis. The two give the same generated topology whenever both apply, and they differ exactly at $\mathcal{S} = \varnothing$ and at families that do not cover: here $\langle \varnothing \rangle$ is the indiscrete topology $\{\varnothing, X\}$, whereas under the other convention it is undefined. Because the choice is invisible in the notation, it is stated in the theorem itself as well as here.

3. "Basis" is a relation, not a property. A family is a basis for a topology; "$\mathcal{B}$ is a basis" alone means "$\mathcal{B}$ is a basis for some topology", and A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis says exactly which families those are and that the topology is then unique. Some texts define a basis abstractly by the two conditions (B1) and (B2) and only afterwards attach a topology to it; others define it only relative to a topology already given, as here. The distinction is harmless once the criterion is available, and it is recorded because the phrase "let $\mathcal{B}$ be a basis" is ambiguous without it. The same remark applies to subbasis, which is always relative to the topology it generates.

4. Coarser and finer, never weaker and stronger. For topologies on one set, $\mathcal{T}_1 \subseteq \mathcal{T}_2$ is read "$\mathcal{T}_1$ is coarser, $\mathcal{T}_2$ is finer" (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). The synonyms smaller/larger are unambiguous and are occasionally used. The pair weaker/stronger is used in both directions in the literature — some authors call the topology with fewer open sets weaker, others call it stronger because it makes more maps continuous into the space — and this library therefore does not use it at all. The discrete topology is the finest and the indiscrete the coarsest (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

Two conventions inherited from earlier pages, which are not forks decided here. They are listed because they change the reading of statements on this page, not because this page chooses them.

• $\mathbb{N}$ contains $0$ and sequences are indexed from $0$ (The natural numbers $\mathbb{N}$ (von Neumann), Sequences of reals: bounded, eventually, frequently, tails, subsequences, Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure). An index range copied from a text that starts at $1$ must be shifted before it is used here; on these pages every radius written $1/(k+1)$ rather than $1/k$ is an instance.
• "Countable" means "at most countable" (Finite, countably infinite, countable, uncountable), so a finite set is countable. Two consequences on this page: a first countable space is allowed a finite neighbourhood base (First countable space: a countable neighbourhood base at every point), which is what makes every discrete space first countable; and the closed sets of the cocountable topology include all the finite sets (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

One thing this page deliberately does not fix. No separation axiom is built into the word space: points need not be closed and distinct points need not be separated by disjoint open sets (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). Older texts, and Bourbaki for compact, build separation into the basic vocabulary; here every separation property is a hypothesis, written out where it is used, and the only one that appears on this page is the Hausdorff condition, quoted from the metric development rather than defined.