Hi dear community,
I have a question on the subject of topology, namely the connectedness of a set.
A set X is called connected if there are no 2 open, nonempty subsets with
Now my problem is the following:
To which topology does the term "open" refer in this definition?
Consider, for example, the interval (0,2) .
Then I can define the topology T(I) of open subsets for the interval I = (0,2) .
In fact, this does give a topology, since
and so on.
But then I = ,
with (0,1] and (1,2) open in I. This would mean that I is NOT connected.
However, using the trivial topology , one finds that I is in fact connected.
The same would be true for itself, e.g. using the trivial topology, the real numbers are connected,
but using the discrete topology, so that every set in (the power set of the real numbers) is open,
then is indeed NOT connected!
However, in many math books it is mentioned that any interval I is connected, but it's not referred to the topology used.
So is connectedness of a set really dependent on the topology used, or what the heck is my conceptual or logical error?
Thanks in advance for help!