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?
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 B.
We have a continuous path g:I->X with I=
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.
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?