In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and other branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' is
locally connected if every point admits a
neighbourhood basis consisting entirely of
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
,
connected sets.
Background
Throughout the history of topology,
connectedness and
compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
have been two of the most
widely studied topological properties. Indeed, the study of these properties even among subsets of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, 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
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
, 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
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s, 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
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
of manifolds is relatively simple (as manifolds are essentially
metrizable according to most definitions of the concept), their
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
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
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
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 In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
is totally disconnected but not
discrete.
Definitions
Let
be a topological space, and let
be a point of
A space
is called locally connected at
[Munkres, p. 161] if every
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of
contains a
connected ''open'' neighborhood of
, that is, if the point
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
for example); and connectedness does not imply local connectedness (see the
topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the functi ...
).
A space
is called locally path connected at
if every neighborhood of
contains a
path connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
''open'' neighborhood of
, that is, if the point
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 In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square ''S'', i.e. on the set of points (''x'',''y'') in the plane such that and
Construction
The ...
).
Connectedness im kleinen
A space
is called connected im kleinen at
or weakly locally connected at
if every neighborhood of
contains a connected neighborhood of
, that is, if the point
has a neighborhood base consisting of connected sets. A space is called weakly 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
is connected im kleinen at
The converse does not hold, as shown for example by a certain infinite union of decreasing
broom space
In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also ...
s, that is connected im kleinen at a particular point, but not locally connected at that point.
[Steen & Seebach, example 119.4, p. 139][Munkres, exercise 7, p. 162] However, if a space is connected im kleinen at each of its points, it is locally connected.
[Willard, Theorem 27.16, p. 201]
A space
is said to be path connected im kleinen at
[, section 2] if every neighborhood of
contains a path connected neighborhood of
, that is, if the point
has a neighborhood base consisting of path connected sets.
A space that is locally path connected at
is path connected im kleinen at
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.
First examples
# For any positive integer ''n'', the Euclidean space
is locally path connected, thus locally connected; it is also connected.
# More generally, every
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
is locally connected, since each point has a local base of
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
(and hence connected) neighborhoods.
# The subspace