Page 117 - DMTH503_TOPOLOGY

Unit 11: Connected Spaces, Connected Subspaces of Real Line

Notes
Example 3: Show that every indiscrete space is connected.
Solution: Let (X, T) be an indiscrete space so that T = {, X}. Then T-open sets are , X. T-closed sets
are X, . Hence the only non-empty subset of X which is both open and closed is X.
 X is connected, by theorem (4).

Self Assessment

1. Prove that the closure of connected set is connected.

2. Prove that a continuous image of a connected space is a connected set.
3. Prove that connectedness is preserved under continuous map.

11.2 Connected Subspaces of Real Line

Theorem 6: The set of real numbers with the usual metric is a connected space.
Proof: Let if possible (R, U) be a disconnected space. Then there most exist non-empty closed
subsets A and B of R such that
A B = R and A B = 
Since A and B are non-empty,  a point a  A and b  B
Since A B = 
 a  b

Thus a < b or a > b
Let a < b
We have [a, b]  p
 [a, b]  A B

Thus x  [a, b]  x  A or x  B
Let p = sup([a, b] A)
Then a  p  b
Since A is closed, p  A
Again A B =  and p  B
 p < b

Also by definition of p
p +   B   > 0
 p +   b

Again since B is closed, p  B.
Thus, we get
p  A and p  B  p  A B
But A B = 
Thus we get a contradiction. Hence R is connected.

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