Locally connected space
In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
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 (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 weak local connectedness at a point and its relation to local connectedness will be considered later on in the article.
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