Page 126 - DMTH503_TOPOLOGY

Topology

Notes
Example 2: Each interval and each ray in the real line in both connected and locally
connected. The subspace [–1, 0)  (0, 1] of is not connected, but it is locally connected.

12.2.2 Locally Connected Subset

Let (X, T) be a topological space and let (Y, T ) be a sub-space of (X, T)
y
The subset Y X is said to be locally connected if (Y, T ) is a locally connected space.
y
12.2.3 Theorems and Solved Examples

Theorem 7: Every discrete space is locally connected.
Solution: Let x be an arbitrary point of a discrete space X. We know that every subset of a
discrete space is open and that every singleton set is connected. Hence {x} is a connected open
nhd. of x. Also every open nhd. of x must contain {x}.
Hence X is locally connected.

Example 3: Give two examples of locally connected space which are not connected.
Or

Is locally connected space always connected? Justify.
Solution:
1. Let X be a discrete space containing more than one point.
Let x X. Then {x} is an open connected set and is obtained in every open set containing x.
So, X is locally connected at each point of X. Also, every singleton subset of X is a non-empty
proper subset of X which is both open and closed. So X is disconnected.
2. Consider the usually topological space (R, U)
Let A R, which is the union of two disjoint open intervals.
Then A is not a interval and therefore it is not connected.
To show that A is locally connected.

Let x be an arbitrary point of A and G be a set open in A such that x G . Then there exists
x x
an open interval I such that x  I G . But I being an interval, it is connected in R and
x x x x
therefore in A.
Thus every open nhd. of x in A contains a connected open nhd. of x in A.
Hence A is locally connected.

Example 4: Give example of a space which is connected but not locally connected.

Solution: Consider the subspace A B of the Euclidean Plane R , where
2
A = {(0, y) : –1 y 1}
  1  
and B =  (x,y) : y  sin   ,0 x  
1

  x  
The A B = and each point of A is a limit point of B and so A and B are not separated.
Consequently, A B is connected.

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