Everything about Connected Space totally explained
In
topology and related branches of
mathematics, a
connected space is a
topological space which can't be represented as the
disjoint union of two or more nonempty
open subsets. Connectedness is one of the principal
topological properties that's used to distinguish topological spaces. A stronger notion is that of a
path-connected space, which is a space where any two points can be joined by a
path.
A subset of a topological space
X is a
connected set if it's a connected space when viewed as a
subspace of
X.
It is usually easy to think about what isn't connected. A simple example would be a space consisting of two
rectangles, each of which is a space and not adjoined to the other. The space isn't connected since two rectangles are disjoint. Another good example is a space with an
annulus removed. The space isn't connected since you can't connect two points, one inside the annulus and the other outside; hence the term "connect".
Formal definition
A
topological space X is said to be
disconnected if it's the
union of two
disjoint nonempty open sets. Otherwise,
X is said to be
connected. A
subset of a topological space is said to be connected if it's connected under its
subspace topology. Some authors specifically exclude the
empty set with its unique topology as a connected space, but this encyclopedia doesn't follow that practice.
For a topological space
X the following conditions are equivalent:
- X is connected.
- X can't be divided into two disjoint nonempty closed sets.
- The only sets which are both open and closed (clopen sets) are X and the empty set.
- The only sets with empty boundary are X and the empty set.
- X can't be written as the union of two nonempty separated sets.
The
maximal connected subsets of any topological space are called the
connected components of the space.
The components form a
partition of the space (that is, they're
disjoint and their union is the whole space).
Every component is a
closed subset of the original space.
The components in general need not be open: the components of the
rational numbers, for instance, are the one-point sets.
A space in which all components are one-point sets is called
totally disconnected. Related to this property, a space
X is called
totally separated if, for any two elements
x and
y of
X, there exist disjoint
open neighborhoods
U of
x and
V of
y such that
X is the union of
U and
V. Clearly any totally separated space is totally disconnected, but the converse doesn't hold. For example take two copies of the rational numbers
Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space isn't totally separated. In fact, it isn't even
Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.
Examples
The closed interval [0, 2] in the standard subspace topology is connected; although it can, for example, be written as the union of [0,1) and [1, 2], the second set isn't open in the aforementioned topology of [0,2].
The union of [0, 1)and (1, 2] is disconnected; both of these intervals are open in the standard topological space [0, 1) ∪ (1, 2].
(0, 1) ∪ is a nonempty family of connected subsets of a topological space X such that is nonempty, then is also connected.
Every path-connected space is connected.
Every locally path-connected space is locally connected.
A locally path-connected space is path-connected iff it's connected.
The closure of a connected subset is connected.
The connected components are always closed (but in general not open)
The connected components of a locally connected space are also open.
The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
Every quotient of a connected (resp. path-connected) space is connected (resp. path-connected).
Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
Every manifold is locally path-connected.Further Information
Get more info on 'Connected Space'.
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