Page 125 - DMTH503_TOPOLOGY

Unit 12: Components and Local Connectedness

Further a  H, H connected and E a component of a implies that H is a subset of E. But G H , Notes
so that G contains a point of H and hence of E.
Thus we have shown that every basic nhd of x contains a point of E.
Consequently every nhd of x will contain a point of E and therefore x E.


Thus x X  x E  E, so that X  E But E  X.
Hence X = E and is therefore connected.
Theorem 6: The component of a topological space X form a position of X i.e. any two components
of X are either disjoint or identical and the union of all the components is X.
Proof: For each x  X, let C (X, x) the union of all connected sets containing x.
Then C (X, x) is a component of X.
Clearly, the family {C : x X} consists of all components of X and X =  {C : x X}. Now let
x x
C (X, x ) and C (X, x ) be the components of X with respect of x and x respectively, x x
1 2 1 2 1 2
If C(X, x ) C (X, x ) = , we are done
1 2
so, let C(X, x )  C (X, x ) = 
1 2
Let x C (X, x )  C (X, x )
1 2
them x  C (X, x ) and x C (X, x )
1 2
Now C (X, x ) and C (X, x ) are connected sets containing x and C (X, x) is a component containing
1 2
x, therefore
C (X, x ) C (X, x)
1
and C (X, x ) (X, x)
2
But C (X, x ) and C (X, x ) being components, they cannot be contained in a larger connected
1 2
subset of X.
Therefore C (X, x ) = C (X, x ) = C (X, x)
1 2
Thus, any two components of X one either disjoint or identical.

Hence, the components of X form a partition of X.
Self Assessment

1. Prove that the components of E corresponding to different points of E are either equal or
disjoint.

12.2 Local Connectedness

12.2.1 Locally Connected Spaces

A topological space X is said to locally connected at a point x X if every nhd. of x contains a
connected nhd. of x i.e. if N is any open set containing x then there exists a connected open set G
containing x such that G  N
or
A topological space (X, T) is said to be locally connected iff for every point x  X and every nhd.
G of x, there exists a connected nhd. H such that x  H G. Thus the space (X, T) is locally
connected iff the family of all open connected sets is a base for T.

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