Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point

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) and let $x\in X$. Subsets carry the subspace topology (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); connectedness is Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets and path-connectedness is Paths, path-connected spaces and path components.

• $X$ is locally connected at $x$ when for every open $U$ with $x\in U$ there is an open connected $V$ with $x\in V\subseteq U$.
• $X$ is locally connected when it is locally connected at every point.
• $X$ is locally path-connected at $x$ when for every open $U$ with $x\in U$ there is an open path-connected $V$ with $x\in V\subseteq U$; and locally path-connected when this holds at every point.

The neighbourhood-base reading. $X$ is locally connected at $x$ exactly when the open connected sets containing $x$ form a neighbourhood base at $x$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open). Indeed a neighbourhood $N$ of $x$ contains an open $U$ with $x\in U\subseteq N$, and an open connected $V$ with $x\in V\subseteq U$ is then a member of that family inside $N$; conversely a base member inside an open $U\ni x$ is exactly what the displayed condition asks. The same sentence with "path-connected" in place of "connected" gives the reading for local path-connectedness. Recall that in this library a neighbourhood need not be open (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open), which is why "open" is written out in both clauses above.

Openness in the clauses is not removable and is a live fork. Asking only for a connected neighbourhood inside every open $U\ni x$ — with no openness demanded of the connected set — defines an a priori weaker condition at a single point, called connectedness im kleinen at $x$ in the literature. This library takes the definition above, with openness, and no statement here asserts that the two agree, at a point or globally.

Local and global connectedness are independent conditions, and neither clause above mentions the other. A two-point discrete space (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) is locally connected, every singleton being open and connected, and is not connected, the two singletons separating it. So local connectedness does not imply connectedness. The reverse implication is not asserted here either.

Both notions are properties of the space, not of an ambient pair. "A locally connected subset $A\subseteq X$" means that the space $A$ with its subspace topology is locally connected, and the open sets tested are then the sets open in $A$.

• Why the notion is stated at a point and then quantified. Almost every application needs the pointwise form: a space can fail to be locally connected at a single point and be perfectly well behaved everywhere else, and naming the bad point is what a counterexample does. Quantifying afterwards costs one line and keeps both forms available.

• The relation to components. The condition says that arbitrarily small open connected sets exist around each point. Since the component of $x$ inside an open $U$ is the largest connected subset of $U$ containing $x$ (Connected components, quasicomponents, and totally disconnected spaces), the definition is asking that those components be large enough to be neighbourhoods — which is exactly the reformulation proved as the next item on this page.

• Local path-connectedness is strictly the stronger-looking of the two, and nothing here compares them. Every path-connected space is connected, so an open path-connected set is an open connected set and local path-connectedness implies local connectedness once that implication is available; it is proved later on this page and is not assumed in this definition.