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 $\displaystyle A,B \subset X $ with

$\displaystyle A \cup B = X $.

Now my problem is the following:

To which topology does the term "open" refer in this definition?

Consider, for example, the interval (0,2) $\displaystyle \subset \mathbb{R} $.

Then I can define the topology T(I) $\displaystyle := \{I, \emptyset, (0,1], (1,2)\} $ of open subsets for the interval I = (0,2) .

In fact, this does give a topology, since $\displaystyle (0,1] \cup (1,2) = I, \\ (0,1] \cap (1,2) = \emptyset $

and so on.

But then I = $\displaystyle (0,1] \cup (1,2) $,

with (0,1] and (1,2) open in I. This would mean that I is NOT connected.

However, using the trivial topology $\displaystyle T'(I) := \{I, \emptyset\} $, one finds that I is in fact connected.

The same would be true for $\displaystyle \mathbb{R} $ itself, e.g. using the trivial topology, the real numbers are connected,

but using the discrete topology, so that every set in $\displaystyle \mathcal{P}(\mathbb{R}) $ (the power set of the real numbers) is open,

then $\displaystyle \mathbb{R} $ 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!