Page 128 - DMTH503_TOPOLOGY

Topology

Notes A = {(x, y) : x = 0, y  [–1, 1]}
1
and B = {(x, y) : 0  x  1 and y = sin }.
x
Since B is the image of (0, 1] under a continuous mapping f give by
 1 

f (x) = x,sin 
 x 
So B is connected.
( Continuous image of a connected space is connected).

Since X = B , therefore X is connected. But it is not locally connected because each point x  A has
a nhd. which does not contain any connected nhd. of x.
Theorem 11: The image of a locally connected space under a mapping which is both open and
continuous is locally connected. Hence locally connectedness is a topological property.
Proof: Let X be a locally connected space and Y be an arbitrary topological space.
Let f : X  Y be a map which is both open and continuous. Without any loss of generality we may
assume that f is onto. We shall show that Y = f (X) is locally connected.
Let y = f (x), x  X, be any point of Y and G be any nhd. of y. Since f is continuous.
 f (G) is open in X containing f (y) = x.
–1
–1
–1
Thus, f (G) is open, nhd. of x.
Now X being locally connected, these exists a connected open set H such that x  H  f (G).
–1
 y = f (x)  f (H)  f [f (G)]  G,
–1
where f (H) is open, since f is open.
Moreover, the continuous image of a connected set is connected, it follows that f (H) is connected.
This shows that f (X) is locally connected at each point.
Hence, f (X) is locally connected.

Self Assessment

2. Show that a connected subspace of a locally connected space has a finite number of
components.
3 Show that the product X × Y of locally connected sets X and Y is locally connected.

12.3 Summary

 A subset of E of a topological space X is said to be a component of X if
(i) E is a connected set &
(ii) E is not a proper subset of any connected subspace of X i.e. if E is a maximal connected
subspace of X.

 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.

 Let (X, T) be a topological space and let (Y, T ) be a subspace of (X, T). The subset y  X is
y
said to be locally connected if (y, T ) is a locally connected space.
y

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