limit points and closure for connected sets
The below theorem shows that adding limit points to a connected
set preserves connectedness.
Theorem 1.
Suppose is a connected set in a topological space.
If , then is connected.
In particular, is connected.
Thus, one way to prove that a space is connected is to find a dense subspace in which is connected.
Two touching closed balls in shows that this theorem does not hold
for the interior. Along the same lines, taking the closure
does not
preserve separatedness.
Proof.
Let be the ambient topological space.
By assumption, if are open and , then
.
To prove that is connected, let be open sets in
such that and for a
contradition, suppose that .
Then there are open sets such that
It follows that and . Next, let be open sets in defined as
Now
and as , it follows that . Then, by the properties of the closure operator,
∎
Title | limit points and closure for connected sets |
---|---|
Canonical name | LimitPointsAndClosureForConnectedSets |
Date of creation | 2013-03-22 15:17:56 |
Last modified on | 2013-03-22 15:17:56 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 7 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 54D05 |