In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space X is locally path connected if every point admits a neighbourhood basis consisting of open path connected sets.
Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of compact subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, connected subsets of R n {\displaystyle \mathbb {R} ^{n}} (for n > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). This led to a rich vein of research in the first half of the twentieth century, in which topologists studied the implications between increasingly subtle and complex variations on the notion of a locally connected space. As an example, the notion of connectedness im kleinen at a point and its relation to local connectedness will be considered later on in the article. In the latter part of the twentieth century, research trends shifted to more intense study of spaces like manifolds, which are locally well understood (being locally homeomorphic to Euclidean space) but have complicated global behavior. By this it is meant that although the basic point-set topology of manifolds is relatively simple (as manifolds are essentially metrizable according to most definitions of the concept), their algebraic topology is far more complex. From this modern perspective, the stronger property of local path connectedness turns out to be more important: for instance, in order for a space to admit a universal cover it must be connected and locally path connected. A space is locally connected if and only if for every open set U, the connected components of U (in the subspace topology) are open. It follows, for instance, that a continuous function from a locally connected space to a totally disconnected space must be locally constant. In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general: for instance Cantor space is totally disconnected but not discrete.
Definitions Let X {\displaystyle X} be a topological space, and let x {\displaystyle x} be a point of X . {\displaystyle X.}
A space X {\displaystyle X} is called locally connected at x {\displaystyle x} if every neighborhood of x {\displaystyle x} contains a connected open neighborhood of x {\displaystyle x} , that is, if the point x {\displaystyle x} has a neighborhood base consisting of connected open sets. A locally connected space is a space that is locally connected at each of its points. Local connectedness does not imply connectedness (consider two disjoint open intervals in R {\displaystyle \mathbb {R} } for example); and connectedness does not imply local connectedness (see the topologist's sine curve). A space X {\displaystyle X} is called locally path connected at x {\displaystyle x} if every neighborhood of x {\displaystyle x} contains a path connected open neighborhood of x {\displaystyle x} , that is, if the point x {\displaystyle x} has a neighborhood base consisting of path connected open sets. A locally path connected space is a space that is locally path connected at each of its points. Locally path connected spaces are locally connected. The converse does not hold (see the lexicographic order topology on the unit square).
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