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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. ==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 (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. 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. Local path connectedness will be discussed as well. 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 and first examples== Let ''X'' be a topological space, and let ''x'' be a point of ''X''. We say that ''X'' is locally connected at ''x'' if for every open set ''V'' containing ''x'' there exists a connected, open set ''U'' with . The space ''X'' is said to be locally connected if it is locally connected at ''x'' for all ''x'' in ''X''.〔Willard, Definition 27.4, p. 199〕 Note that local connectedness and connectedness are not related to one another; a space may possess one or both of these properties, or neither. By contrast, we say that ''X'' is weakly locally connected at ''x'' (or connected im kleinen at ''x'') if for every open set ''V'' containing ''x'' there exists a connected subset ''N'' of ''V'' such that ''x'' lies in the interior of ''N''. An equivalent definition is: each open set ''V'' containing ''x'' contains an open neighborhood ''U'' of ''x'' such that any two points in ''U'' lie in some connected subset of ''V''.〔Willard, Definition 27.14, p. 201〕 The space ''X'' is said to be weakly locally connected if it is weakly locally connected at ''x'' for all ''x'' in ''X''. In other words, the only difference between the two definitions is that for local connectedness at ''x'' we require a neighborhood base of ''open'' connected sets containing ''x'', whereas for weak local connectedness at ''x'' we require only a neighborhood base of ''connected'' sets containing ''x''. Evidently a space which is locally connected at ''x'' is weakly locally connected at ''x''. The converse does not hold (a counterexample, the broom space, is given below). On the other hand, it is equally clear that a locally connected space is weakly locally connected, and here it turns out that the converse does hold: a space which is weakly locally connected at all of its points is necessarily locally connected at all of its points.〔Willard, Theorem 27.16, p. 201〕 A proof is given below. We say that ''X'' is locally path connected at x if for every open set ''V'' containing ''x'' there exists a path connected, open set ''U'' with . The space ''X'' is said to be locally path connected if it is locally path connected at ''x'' for all ''x'' in ''X''. Since path connected spaces are connected, locally path connected spaces are locally connected. This time the converse does not hold (see example 6 below). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「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.==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 \mathbb^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 weak local connectedness 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. Local path connectedness will be discussed as well. 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 and first examples== Let ''X'' be a topological space, and let ''x'' be a point of ''X''.We say that ''X'' is locally connected at ''x'' if for every open set ''V'' containing ''x'' there exists a connected, open set ''U'' with x \in U \subset V. The space ''X'' is said to be locally connected if it is locally connected at ''x'' for all ''x'' in ''X''.Willard, Definition 27.4, p. 199 Note that local connectedness and connectedness are not related to one another; a space may possess one or both of these properties, or neither.By contrast, we say that ''X'' is weakly locally connected at ''x'' (or connected im kleinen at ''x'') if for every open set ''V'' containing ''x'' there exists a connected subset ''N'' of ''V'' such that ''x'' lies in the interior of ''N''. An equivalent definition is: each open set ''V'' containing ''x'' contains an open neighborhood ''U'' of ''x'' such that any two points in ''U'' lie in some connected subset of ''V''.Willard, Definition 27.14, p. 201 The space ''X'' is said to be weakly locally connected if it is weakly locally connected at ''x'' for all ''x'' in ''X''.In other words, the only difference between the two definitions is that for local connectedness at ''x'' we require a neighborhood base of ''open'' connected sets containing ''x'', whereas for weak local connectedness at ''x'' we require only a neighborhood base of ''connected'' sets containing ''x''.Evidently a space which is locally connected at ''x'' is weakly locally connected at ''x''. The converse does not hold (a counterexample, the broom space, is given below). On the other hand, it is equally clear that a locally connected space is weakly locally connected, and here it turns out that the converse does hold: a space which is weakly locally connected at all of its points is necessarily locally connected at all of its points.Willard, Theorem 27.16, p. 201 A proof is given below.We say that ''X'' is locally path connected at x if for every open set ''V'' containing ''x'' there exists a path connected, open set ''U'' with x \in U \subset V. The space ''X'' is said to be locally path connected if it is locally path connected at ''x'' for all ''x'' in ''X''.Since path connected spaces are connected, locally path connected spaces are locally connected. This time the converse does not hold (see example 6 below).」の詳細全文を読む スポンサード リンク
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