Connected Sets in Topology

Date: 04/22/98 at 11:13:49
From: DeAndrea Flowers
Subject: Topology

I know that this is supposed to be for high school level math, but I 
am really stuck with this homework assignment. Can you please help me?

The question states: 

     Let X be a topological space with x an element of X. Show that 
     {x} is connected.
 
I really have no idea of how to go about showing this. 

Another problem that I am having trouble solving is:

     Give an example to show that the union of two connected sets is      
     not necessarily connected.

The only thing that I know is the definition of connected: 

     A topological space X is said to be disconnected if there are 
     disjoint nonempty open subsets U and V of X for which 
     X=(U and V). If it is not disconnected then it is connected.

Can you please try to give me some advice on how to solve these 
problems, or maybe suggest another Web site that might be able to 
help? Thank you.

Date: 04/22/98 at 14:19:23
From: Doctor Tom
Subject: Re: Topology

Hi DeAndrea,

If you're a beginner in topology (which I guess you are), the best way
to think about these problems is in terms of some specific examples.

I have found that for the elementary stuff, the best specific example 
is just the usual two-dimensional plane or the one-dimensional 
Euclidean line, or subsets of them, with the usual definition of open 
sets in Euclidean space.

"Connected" means pretty much what you think it should. If I look at
two circles of the plane that don't intersect, they are disconnected,
in the sense that there's a gap between them - you can't get from
one to the other and you can split the space up into the two circles, 
call them U and V, and you will see that this division makes up the 
whole space, so it's disconnected.

But look at one circle. You think of it as connected, right?  Well,
try to imagine dividing it into two open sets. You can hack off an 
open chunk, but you'll always find that what's left isn't open - it 
includes the boundary, so the other part is not open.

If this is confusing, just think of one dimension. If I have two open 
intervals: (0,1) and (2,3), they're clearly disconnected, they're both 
open, and will satisfy the U and V of the definition of disconnected 
above. But try to hack (0,1) into two open sets. If you take one piece 
as (0, 1/2), the other piece will be [1/2,1). Note that the number 1/2 
is in the second chunk, so it's not open. This will happen no matter 
how you try to divide it.

Your original question is easy - since there's only one element in 
{x}, you CAN'T divide it into two non-empty sets - one will have one 
element and the other zero, so one of the sets will have to be empty.  
Therefore, it can't be disconnected, so it must be connected.

-Doctor Tom,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/