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
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.