# (Path) connected spaces

• Oct 28th 2010, 11:09 AM
Sogan
(Path) connected spaces
Hello,

i have a question about connected and path connected spaces.

in my book i have read, that every path-connected space is also connected.
It is proved there as follows:

Suppose that X isn't connected then there exist nonempty-open sets A,B s.t
X=A \$\displaystyle \cup \$ B.

We have a continuous path g:I->X with I=\$\displaystyle g^-1(A) \cup g^-1(B)\$

But I is connected, so here is a contradiction.

I want to know, why this proof is correct? I mean I is sometimes open sometimes not determined by the topology given on I. So how can we say I is open.
I think the author means the intervall I with the standard-topology given by |.|.

But why does this proof works? I'm a little bit confused.

I hope someone can explain this problem for me.
• Oct 28th 2010, 11:41 AM
Plato
I don't know what the author means from that brief description.
However, there is a point in A and a point in B because they are not empty.
But there is a path between the two points.
If A is separated from B, then how can that happen?
• Oct 28th 2010, 11:49 AM
Sogan
Yes of course this must be true. The intuition is so clear. But i dont understand the mathematic proof given there.
I would also be happy with other proofs, which i can understand.
• Oct 28th 2010, 11:55 AM
Plato
Quote:

Originally Posted by Sogan
Yes of course this must be true. The intuition is so clear. But i dont understand the mathematic proof given there.
I would also be happy with other proofs, which i can understand.

Every path is a connect subset of the space.
If \$\displaystyle A\cup B\$ is a dis-connection of the space then any path must either be a subset of \$\displaystyle A\$ or \$\displaystyle B\$.
That is a standard theorem proven in a in topology.
• Oct 28th 2010, 12:04 PM
Sogan
OK, thanks for your help. Excuse me, i'm not so familiar with topological concepts. This is new for me.
I have just one question to your argumentation: What do you mean by "path is connect subset"?
is it the image of the path? if yes, why it is connect?
• Oct 28th 2010, 12:15 PM
Plato
Quote:

Originally Posted by Sogan
OK, thanks for your help. Excuse me, i'm not so familiar with topological concepts. This is new for me.

The pray tell why are working with topological concepts?