Disconnected Space
A topological space X is said to be disconnected space if X can be separated as the union of two nonempty disjoint open sets.
In other words, a topological space X is said to be a disconnected space if there exist non-empty
open sets A andB such that
of X.
and
. The pair
is called the disconnection
Example:
Show that two point discrete spaces is disconnected.
Let X be a two point discrete space, If A is any proper subset of X, then both
empty open subsets of X such that
and
disconnection of X, so X is a disconnected space.
. This shows that
and
are nonis a
Examples:

is a disconnected.

with upper limit topology is disconnected, since
which form a disconnection of .

Let

Then
is a disconnected space.
Every discrete space with more than one point is disconnected.
be a non-empty set, with topology
and
are both open sets
defined on X.
Connected Space
A topological space which cannot be written as the union of two non-empty disjoint open sets is said
to be a connected space.
In other words, a space X is connected if it is not the union of two non-empty disjoint open sets.
Example:
Every indiscrete space is connected.
Let X be an indiscrete space, then X is the only non-empty open set, so we cannot find the
disconnection of X. Hence X is connected.
Connected Subspace:
A subspace Y of a topological space is said to connected subspace if Y is a connected as a
topological space in its own right.
Example:
The subset of a connected space may not be connected.
The set of real numbers
cannot be represented as the union of two disjoint non-empty sets, so
is a connected space.
Next suppose that
and
Since
subsets of
and
. Also
and
are open subsets of
. This shows that
, so A and B are open
is a disconnected subspace of
.
Theorems:





A topological space is connected if and only if cannot be represented as the union of two disjoint
non-empty closed sets.
An infinite set with co-finite topology is a connected space.
Any continuous image of a connected space is connected.
The range of a continuous real unction defined on a connected space is an interval.
If
is a disconnection of X and C is a connected subspace of X, Then C is contained either
in A or in B.
Characterization of Connected Space:
In a space, the following are equivalent:
o
X is connected.
o
The only open and closed subsets of X are
o
There does not exist a continuous map
space
.
, X.
from a space X onto the discrete
Components of a Space
A connected subspace of a topological space X is said to be the component of X if it is not properly
contained in any connected subspace of X.
Note: The component of a connected space X is the whole space, X, itself.
Example:
The singleton subset of a two-point discrete space is its components.
Let X is be two-point discrete space, then the only possible connected subsets of X are its singleton
subsets. As no singleton subset is properly contained in any other singleton subset, so these singleton
subsets are components of X. Hence the singleton subsets of a two-point discrete space are its components.
Theorems:


Let X be a topological space, then for each
, there is exactly one component of Xcontaining x.
Let X be a topological space, then each connected subset of X is contained in a component of X.
Theorem: Let X be a topological space, then every component is closed in X.
Proof: Let
be a component of X. If possible, suppose that
connected and contains
is closed.
is not closed in X. Since
, which is a contradiction to the fact that
is closed and
is a component. Hence
, so
Totally Disconnected Space
A topological space X is said to be totally disconnected space if any pair of distinct of X can be
separated by a disconnection of X.
In other words, a topological space X is said to be totally disconnected space if for any two
points x and y of X, there is a disconnection
of X such that
and
.
In other words, a topological space X is said to be totally disconnected space if its connected subsets
are only the singleton subset of X.
Example: Every discrete space is totally disconnected.
Let X be a discrete space. Let
subsets of X such that
, then
. Since
and
are open
is a disconnection of X, so X is totally
disconnected.
Examples:

One point space is totally disconnected.




is totally disconnected.
is totally disconnected.
The Cantor set is totally disconnected.
with usual topology is not totally disconnected.

with upper limit topology generated by open-closed intervals
is totally disconnected.
Theorems:



Every totally disconnected space is Hausdorff space.
The components of a totally disconnected space are its singleton subsets.
If a Hausdorff space X has an open base whose sets are also closed then X is totally disconnected.
Some Applications of Connectedness
In some applications of connectedness, we shall define two fixed point theorems in connection with
application of connectedness. Fixed point theorems are useful in obtaining the (unique) solutions of
differential and integral equations.
Fixed Point:
Let
be a self mapping. A point
may be noted that not every mapping has a fixed point.
is called a fixed point of
, if
Examples:

Let
be defined by

Let
be defined by

Let
be defined by
. Then each point
. Then
. Then
is a fixed point.
has no fixed point.
has exactly one fixed point “0”.
. It
Theorems:

Let

that
.
A contraction self mapping T defined on a complete metric space X has a unique fixed point.
be a continuous self mapping. Then there exist a point
such
Fixed Point Space:
A space X is said to be a fixed point space, if every continuous self mapping on X has a fixed point.
For example,
is a fixed point space or in other words, we say that X has a fixed point property.
Finally, we show that “a fixed point property” is a topological property.
Theorem:
Let X and Y be homeomorphic spaces. Then each continuous mapping
point if and only if each continuous mapping
has a fixed
has a fixed point.
Compact Space
Cover and Sub-Cover:
Let X be a topological space. A collection
of X if
of subsets of X is said to be a cover
.
A sub-collection of
is said to be a sub-cover of X if itself is a cover of X.
Open Cover and Open Sub-Cover:
Let X be a topological space. A collection
cover of X if
of open subsets of X is said to be an open
.
A sub-collection of
is said to be a sub-cover of X if itself is an open cover ofX.
Compact Space:
A compact space is a topological space in which every open cover has a finite sub-cover.
Compact Subspace:
A compact subspace of a topological space is a subspace which is compact as a topological space in
its own right.
Examples:

Every finite topological space is compact.

Let
be a topological space where
compact space.
consists of finite number of elements, then X is a
Theorems:






An infinite set with co-finite topology is a compact space.
The real line
is not compact.
Every closed subspace of a compact space is compact.
The continuous image of compact space is compact.
The homeomorphic image of compact space is compact.
Any continuous bijective function from a compact space X is a hausdorff space Y is a
homeomorphism.
Finite Intersection Property
A collection A of subsets of a non-empty set X is said to have the finite intersection property if every
finite sub-collection of A has non-empty intersection.
In other words, the collection
of subsets of the topological space X is said to
have finite intersection property if every finite sub-collection of A has non-empty intersection, i.e. for any
finite subset
Theorem:
of
,
.
A topological space X is compact if and only if every collection
of X which satisfies the finite intersection property itself has a non-empty intersection.
of closed sets
Local Compact
A space X is said to be locally compact (briefly L-Compact) at
if and only if x has a compact
neighbourhood in X. If X is L-compact at every point, then Xis called a locally compact space.
Examples:

Compact spaces are L-compact. Suppose X is compact,X is a neighbourhood of each of its points
implies X isL-compact.

The usual real line
is L-compact, since for each
, we have
.
Thus
is a neighbourhood of x which is compact by Heine-Boral theorem. This proves that
is L-compact. But recall that
is not compact.

and
as subspace of
are not locally compact.
Theorems:
o
o
A compact space is L-compact.
If X is a Hausdorff locally compact space, then for all
and for all
neighbourhoods U of x, there exists a compact neighbourhood V of x such that
o
o
o
.
Let
be an open continuous surjection. If X is L-compact, the Y is L-compact.
Local compactness is a closed hereditary property.
are L-compact if and only if
is L-compact.