Intopology and other branches ofmathematics, atopological spaceX islocally connected if every point admits aneighbourhood basis consisting ofopenconnected sets.

As a stronger notion, the spaceX islocally path connected if every point admits a neighbourhood basis consisting of openpath connected sets.

Throughout the history of topology,connectedness andcompactness have been two of the mostwidely studied topological properties. Indeed, the study of these properties even among subsets ofEuclidean space, and the recognition of their independence from the particular form of theEuclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure ofcompact subsets of Euclidean space was understood quite early on via theHeine–Borel theorem,connected subsets of${\displaystyle {\mathbb{R}}^n}$ (forn > 1) proved to be much more complicated. Indeed, while any compactHausdorff space islocally 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 ofconnectedness 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 likemanifolds, which are locally well understood (beinglocally homeomorphic to Euclidean space) but have complicated global behavior. By this it is meant that although the basicpoint-set topology of manifolds is relatively simple (as manifolds are essentiallymetrizable according to most definitions of the concept), theiralgebraic 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 auniversal cover it must be connected and locally path connected.

A space is locally connected if and only if for every open setU, the connected components ofU (in thesubspace topology) are open. It follows, for instance, that a continuous function from a locally connected space to atotally 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 instanceCantor space is totally disconnected but notdiscrete.

Let${\displaystyle X}$ be a topological space, and let${\displaystyle x}$ be a point of${\displaystyle X.}$

A space${\displaystyle X}$ is calledlocally connected at${\displaystyle x}$^[1] if everyneighborhood of${\displaystyle x}$ contains aconnectedopen neighborhood of${\displaystyle x}$, that is, if the point${\displaystyle x}$ has aneighborhood base consisting of connected open sets. Alocally connected space^[2][1] is a space that is locally connected at each of its points.

Local connectedness does not imply connectedness (consider two disjoint open intervals in${\displaystyle \mathbb{R}}$ for example); and connectedness does not imply local connectedness (see thetopologist's sine curve).

A space${\displaystyle X}$ is calledlocally path connected at${\displaystyle x}$^[1] if every neighborhood of${\displaystyle x}$ contains apath connectedopen neighborhood of${\displaystyle x}$, that is, if the point${\displaystyle x}$ has a neighborhood base consisting of path connected open sets. Alocally path connected space^[3][1] 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 thelexicographic order topology on the unit square).

A space${\displaystyle X}$ is calledconnected im kleinen at${\displaystyle x}$^[4][5] orweakly locally connected at${\displaystyle x}$^[6] if every neighborhood of${\displaystyle x}$ contains a connected (not necessarily open) neighborhood of${\displaystyle x}$, that is, if the point${\displaystyle x}$ has a neighborhood base consisting of connected sets. A space is calledweakly locally connected if it is weakly locally connected at each of its points; as indicated below, this concept is in fact the same as being locally connected.

A space that is locally connected at${\displaystyle x}$ is connected im kleinen at${\displaystyle x.}$ The converse does not hold, as shown for example by a certain infinite union of decreasingbroom spaces, that is connected im kleinen at a particular point, but not locally connected at that point.^[7][8][9] However, if a space is connected im kleinen at each of its points, it is locally connected.^[10]

A space${\displaystyle X}$ is said to bepath connected im kleinen at${\displaystyle x}$^[5] if every neighborhood of${\displaystyle x}$ contains a path connected (not necessarily open) neighborhood of${\displaystyle x}$, that is, if the point${\displaystyle x}$ has a neighborhood base consisting of path connected sets.

A space that is locally path connected at${\displaystyle x}$ is path connected im kleinen at${\displaystyle x.}$ The converse does not hold, as shown by the same infinite union of decreasing broom spaces as above. However, if a space is path connected im kleinen at each of its points, it is locally path connected.^{[11][better source needed]}

1. For any positive integern, the Euclidean space${\displaystyle {\mathbb{R}}^n}$ is locally path connected, thus locally connected; it is also connected.
2. More generally, everylocally convex topological vector space is locally connected, since each point has a local base ofconvex (and hence connected) neighborhoods.
3. The subspace${\displaystyle S=[0,1]\cup [2,3]}$ of the real line${\displaystyle {\mathbb{R}}^1}$ is locally path connected but not connected.
4. Thetopologist's sine curve is a subspace of the Euclidean plane that is connected, but not locally connected.^[12]
5. The space${\displaystyle \mathbb{Q}}$ ofrational numbers endowed with the standard Euclidean topology, is neither connected nor locally connected.
6. Thecomb space is path connected but not locally path connected, and not even locally connected.
7. A countably infinite set endowed with thecofinite topology is locally connected (indeed,hyperconnected) but not locally path connected.^[13]
8. Thelexicographic order topology on the unit square is connected and locally connected, but not path connected, nor locally path connected.^[14]
9. TheKirch space is connected and locally connected, but not path connected, and not path connected im kleinen at any point. It is in facttotally path disconnected.

Afirst-countableHausdorff space${\displaystyle (X,\tau )}$ is locally path-connected if and only if${\displaystyle \tau }$ is equal to thefinal topology on${\displaystyle X}$ induced by the set${\displaystyle C([0,1];X)}$ of all continuous paths${\displaystyle [0,1]\to (X,\tau ).}$

1. Local connectedness is, by definition, alocal property of topological spaces, i.e., a topological propertyP such that a spaceX possesses propertyP if and only if each pointx inX admits a neighborhood base of sets that have propertyP. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
2. A space is locally connected if and only if it admits abase of (open) connected subsets.
3. Thedisjoint union${\displaystyle \coprod _i{X}_i}$ of a family${\displaystyle \{X_i\}}$ of spaces is locally connected if and only if each${\displaystyle X_i}$ is locally connected. In particular, since a single point is certainly locally connected, it follows that anydiscrete space is locally connected. On the other hand, a discrete space istotally disconnected, so is connected only if it has at most one point.
4. Conversely, atotally disconnected space is locally connected if and only if it is discrete. This can be used to explain the aforementioned fact that the rational numbers are not locally connected.
5. A nonempty product space${\displaystyle \prod _i{X}_i}$ is locally connected if and only if each${\displaystyle X_i}$ is locally connected and all but finitely many of the${\displaystyle X_i}$ are connected.^[15]
6. Everyhyperconnected space is locally connected, and connected.

The following result follows almost immediately from the definitions but will be quite useful:

Lemma: LetX be a space, and${\displaystyle \{Y_i\}}$ a family of subsets ofX. Suppose that${\displaystyle \bigcap _i{Y}_i}$ is nonempty. Then, if each${\displaystyle Y_i}$ is connected (respectively, path connected) then the union${\displaystyle \bigcup _i{Y}_i}$ is connected (respectively, path connected).^[16]

Now consider two relations on a topological spaceX: for${\displaystyle x,y\in X,}$ write:

${\displaystyle x{\equiv }_{c}y}$ if there is a connected subset ofX containing bothx andy; and

${\displaystyle x{\equiv }_{pc}y}$ if there is a path connected subset ofX containing bothx andy.

Evidently both relations are reflexive and symmetric. Moreover, ifx andy are contained in a connected (respectively, path connected) subsetA andy andz are connected in a connected (respectively, path connected) subsetB, then the Lemma implies that${\displaystyle A\cup B}$ is a connected (respectively, path connected) subset containingx,y andz. Thus each relation is anequivalence relation, and defines a partition ofX intoequivalence classes. We consider these two partitions in turn.

Forx inX, the set${\displaystyle C_x}$ of all pointsy such that${\displaystyle y{\equiv }_{c}x}$ is called theconnected component ofx.^[17] The Lemma implies that${\displaystyle C_x}$ is the unique maximal connected subset ofX containingx.^[18] Since the closure of${\displaystyle C_x}$ is also a connected subset containingx,^[19][20] it follows that${\displaystyle C_x}$ is closed.^[21]

IfX has only finitely many connected components, then each component is the complement of a finite union of closed sets and therefore open. In general, the connected components need not be open, since, e.g., there exist totally disconnected spaces (i.e.,${\displaystyle C_x=\{x\}}$ for all pointsx) that are not discrete, like Cantor space. However, the connected components of a locally connected space are also open, and thus areclopen sets.^[22] It follows that a locally connected spaceX is a topological disjoint union${\displaystyle \coprod C_x}$ of its distinct connected components. Conversely, if for every open subsetU ofX, the connected components ofU are open, thenX admits a base of connected sets and is therefore locally connected.^[23]

Similarlyx inX, the set${\displaystyle P{C}_x}$ of all pointsy such that${\displaystyle y{\equiv }_{pc}x}$ is called thepath component ofx.^[24] As above,${\displaystyle P{C}_x}$ is also the union of all path connected subsets ofX that containx, so by the Lemma is itself path connected. Because path connected sets are connected, we have${\displaystyle P{C}_x\subseteq C_x}$ for all${\displaystyle x\in X.}$

However the closure of a path connected set need not be path connected: for instance, the topologist's sine curve is the closure of the open subsetU consisting of all points(x,sin(x)) withx > 0, andU, being homeomorphic to an interval on the real line, is certainly path connected. Moreover, the path components of the topologist's sine curveC areU, which is open but not closed, and${\displaystyle C\setminus U,}$ which is closed but not open.

A space is locally path connected if and only if for all open subsetsU, the path components ofU are open.^[24] Therefore the path components of a locally path connected space give a partition ofX into pairwise disjoint open sets. It follows that an open connected subspace of a locally path connected space is necessarily path connected.^[25] Moreover, if a space is locally path connected, then it is also locally connected, so for all${\displaystyle x\in X,}$${\displaystyle C_x}$ is connected and open, hence path connected, that is,${\displaystyle C_x=P{C}_x.}$ That is, for a locally path connected space the components and path components coincide.

1. The set${\displaystyle I\times I}$ (where${\displaystyle I=[0,1]}$) in thedictionaryorder topology has exactly one component (because it is connected) but has uncountably many path components. Indeed, any set of the form${\displaystyle \{a\}\times I}$ is a path component for eacha belonging toI.
2. Let${\displaystyle f:\mathbb{R}\to {\mathbb{R}}_{\ell }}$ be a continuous map from${\displaystyle \mathbb{R}}$ to${\displaystyle {\mathbb{R}}_{\ell }}$ (which is${\displaystyle \mathbb{R}}$ in thelower limit topology). Since${\displaystyle \mathbb{R}}$ is connected, and the image of a connected space under a continuous map must be connected, the image of${\displaystyle \mathbb{R}}$ under${\displaystyle f}$ must be connected. Therefore, the image of${\displaystyle \mathbb{R}}$ under${\displaystyle f}$ must be a subset of a component of${\displaystyle {\mathbb{R}}_{\ell }/}$ Since this image is nonempty, the only continuous maps from '${\displaystyle \mathbb{R}}$ to${\displaystyle {\mathbb{R}}_{\ell },}$ are the constant maps. In fact, any continuous map from a connected space to a totally disconnected space must be constant.

LetX be a topological space. We define a third relation onX:${\displaystyle x{\equiv }_{qc}y}$ if there is no separation ofX into open setsA andB such thatx is an element ofA andy is an element ofB. This is an equivalence relation onX and the equivalence class${\displaystyle Q{C}_x}$ containingx is called thequasicomponent ofx.^[18]

${\displaystyle Q{C}_x}$ can also be characterized as the intersection of allclopen subsets ofX that containx.^[18] Accordingly${\displaystyle Q{C}_x}$ is closed; in general it need not be open.

Evidently${\displaystyle C_x\subseteq Q{C}_x}$ for all${\displaystyle x\in X.}$^[18] Overall we have the following containments among path components, components and quasicomponents atx:$${\displaystyle P{C}_x\subseteq C_x\subseteq Q{C}_x.}$$

IfX is locally connected, then, as above,${\displaystyle C_x}$ is a clopen set containingx, so${\displaystyle Q{C}_x\subseteq C_x}$ and thus${\displaystyle Q{C}_x=C_x.}$ Since local path connectedness implies local connectedness, it follows that at all pointsx of a locally path connected space we have$${\displaystyle P{C}_x=C_x=Q{C}_x.}$$

Another class of spaces for which the quasicomponents agree with the components is the class of compact Hausdorff spaces.^[26]

1. An example of a space whose quasicomponents are not equal to its components is a sequence with a double limit point. This space is totally disconnected, but both limit points lie in the same quasicomponent, because any clopen set containing one of them must contain a tail of the sequence, and thus the other point too.
2. The space${\displaystyle (\{0\}\cup \{\frac{1}{n}:n\in {\mathbb{Z}}^{+}\})\times [-1,1]\setminus \{(0,0)\}}$ is locally compact and Hausdorff but the sets${\displaystyle \{0\}\times [-1,0)}$ and${\displaystyle \{0\}\times (0,1]}$ are two different components which lie in the same quasicomponent.
3. TheArens–Fort space is not locally connected, but nevertheless the components and the quasicomponents coincide: indeed${\displaystyle Q{C}_x=C_x=\{x\}}$ for all pointsx.^[27]

• Locally simply connected space
• Semi-locally simply connected
• It is conjectured that the Mandelbrot set is locally connected

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• Coppin, C. A. (1972), "Continuous Functions from a Connected Locally Connected Space into a Connected Space with a Dispersion Point",Proceedings of the American Mathematical Society,32 (2), American Mathematical Society:625–626,doi:10.1090/S0002-9939-1972-0296913-7,JSTOR 2037874. For Hausdorff spaces, it is shown that any continuous function from a connected locally connected space into a connected space with a dispersion point is constant
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• Properties of topological spaces
• General topology

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