Which conventions this page fixes: the empty space and the one-point space, separated sets against disjoint open sets, and what is not developed here

Five convention forks are live in the material of this page, and each is settled here rather than left to the reader. Two further conventions are inherited and change how statements here are read.

1. The empty space is connected, and so is a one-point space. A separation asks for two nonempty disjoint open pieces covering the space (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets), and neither space admits one, so both are connected with no special clause. The competing convention adds "nonempty" to the definition of a connected space, which makes the empty space neither connected nor disconnected. The cost of the choice made here is that the empty set is a connected subset of every space, so "maximal connected subset" must be read as "maximal among the nonempty connected subsets" for the components; that is exactly where Connected components, quasicomponents, and totally disconnected spaces and the maximality clause of the components theorem take care. The benefit is that no theorem on this page needs a nonemptiness hypothesis: unions, closures, continuous images and products are all stated without one.

2. A separation is two disjoint open sets; separated sets are a different condition, and both are used. Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets defines a separation of a space by open sets, and defines $A_1, A_2$ to be separated in $X$ when neither meets the other's closure. Those are not the same demand: separated sets need not be open, and the ambient open sets that witness a separation of a subspace need not be disjoint in $X$ at all — Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets requires them to be disjoint only on the subspace, and "requiring $U \cap V = \varnothing$ outright is a strictly stronger demand and is a different notion". A subspace $A \subseteq X$ is disconnected exactly when $A = A_1 \cup A_2$ with $A_1, A_2$ nonempty and separated in $X$, which is the criterion this library already uses on the real line is the theorem relating them, and it is what lets a computation be done in whichever of the two vocabularies is convenient. The real-line development uses the second (Separated sets, disconnection, and connected subset of $\mathbb{R}$), the general development uses the first, and The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$" is where the two are shown to agree on $\mathbb{R}$ — an identification that is proved, never assumed.

3. Local connectedness demands OPEN connected sets. Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point asks that every open $U$ containing $x$ contain an open connected $V$ with $x \in V \subseteq U$. Asking instead only that $U$ contain a connected $V$ that is a neighbourhood of $x$, without requiring $V$ itself to be open, gives a weaker condition at a point, called connectedness im kleinen in the literature; dropping open outright, so that any connected $V$ with $x \in V \subseteq U$ would serve, asks nothing at all, since the singleton $\{x\}$ always qualifies. This page proves nothing about that weaker condition and asserts no relation between the two. The same fork, with the same resolution, applies to local path-connectedness (Paths, path-connected spaces and path components).

4. Totally disconnected is defined by components, not by quasicomponents. Connected components, quasicomponents, and totally disconnected spaces calls $X$ totally disconnected when every component is a singleton. The condition that every quasicomponent is a singleton is a different property, usually called total separatedness, and by Every quasicomponent is a closed union of components, so each component is contained in a quasicomponent, and the quasicomponents partition the space it is at least as strong. Nothing on this page asserts that the two agree.

5. The order topology is generated by the open rays. The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua takes the rays $L_{<a}$ and $L_{>a}$ as a subbasis, which is what makes the definition work uniformly when $L$ has a least or a greatest element; taking the open intervals alone as a basis would fail there. The same item fixes order-convex, order-dense, the least upper bound property and linear continuum, and records that a subspace of a linearly ordered topological space always means the subspace topology, which agrees with the order topology of the restricted order when the subset is order-convex and is not claimed to agree otherwise.

Two inherited conventions that change how this page is read. A neighbourhood need not be open, so "open connected neighbourhood" is written out in full wherever openness is wanted; and the empty intersection of a subbasis is the whole space, so no covering hypothesis is imposed on the subbasis of rays. These conventions are in force throughout the items above.

Notation used without further comment. $\mathbf{2}$ is the two-point discrete space (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies); $C(x)$, $Q(x)$ and $P(x)$ are the component, the quasicomponent and the path component of $x$; and "interval" applied to a subset of $\mathbb{R}$ is read throughout as "order-convex", the classification of the order-convex subsets into written forms not being available here.