Hi

Do you know an exemple of connected space but path connected in no open subset (except perhaps the whole space) ?

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- Mar 3rd 2009, 01:50 PMclic-claca topological space
Hi

Do you know an exemple of connected space but path connected in no open subset (except perhaps the whole space) ? - Mar 7th 2009, 10:17 AMclic-clac
Could anyone tell me if that's a wrong exemple (but I think it works): in

- Mar 7th 2009, 12:20 PMOpalg
- Mar 7th 2009, 02:40 PMbkarpuz

By the way, can you please give the definitions that you use?

*connected*and*path connected*? - Mar 7th 2009, 02:43 PMbkarpuz
- Mar 8th 2009, 12:41 AMclic-clac
Definitions I use is:

connected iff for all disjoint open subsets in or

is path connected iff between any two points in there is always a path. (a continuous map from to such that and ) - Mar 8th 2009, 01:58 AMbkarpuz
Why not only considering

which has the same nature with the first quadrant of

It is clear that this set is not path connected.

Because the mapping will have discontinuity while matching and for some and , for instance when , and in any open subset you may find such points (may be sketching a graphic would help for this).

In this case, you will always have for some .

About the connectivity, I am not sure for this set. - Mar 8th 2009, 03:20 AMclic-clac
You're right it's not path-connected, but unfortunately it's not connected: you can write this space as a disjoint union of non trivial open sets, for instance:

(this sets are open because they are intersections with open subsets in , and we're working with subspace topology)

But the space I was looking for may be more complicated: it has to be connected, but, instead of not path connected, not path-connected in any open subset (i.e. if you want never locally path-connected), and I'm not sure that ( not path-connected) ( not path-connected in any open subset)... - Mar 8th 2009, 03:39 AMbkarpuz
Okay, please see here Connected and Path Connected

I guess this will help you much, and indicates that your assertion is true.