Confusion with Disconnected sets

[Mr-T]

Hello,

I am having some difficulties understanding why a subset under the usual metric topology of the reals is connected.

How can a set X = (0,1] u (1,2) be connected?

The definition I am using is:

A is disconnected if there exists two open sets G and V and the following three properties hold:

(1) A intersect G ≠ ∅
A intersect V ≠ ∅

(2) A is a proper subset of the union of G and V.

(3) the intersection of G and V is the empty set.

Thanks

 

[micromass]

And you also want $G$ and $V$ to be open. No?

 

[Mr-T]

Yea, that would be more correct.

 

[HallsofIvy]

Okay, so can you find two such open sets for $(0, 1]\cup (1, 2)$? (0, 1] and (1, 2) will not do because (0, 1] is not open. (And, did you notice that $(0, 1]\cup (1, 2)= (0, 2)$?)

 

[blue_raver22]

for your question. It is understandable to have confusion about disconnected sets, as the concept can be counterintuitive at first. However, let me try to explain why the set X = (0,1] u (1,2) is actually connected.

First, let's review the definition of a connected set. A set is connected if it cannot be divided into two non-empty subsets that are both open and disjoint. In other words, there is no way to separate the set into two open sets that do not intersect.

Now, let's look at the set X = (0,1] u (1,2). This set consists of two disjoint open intervals, (0,1] and (1,2), which are connected to each other by the point 1. This means that no matter how we try to divide X into two open sets, at least one of those sets will have to include the point 1, thus making them intersect. Therefore, X cannot be divided into two non-empty, disjoint open sets, and is therefore connected.

I hope this explanation helps clarify the concept of connected sets for you. If you have any further questions or need more clarification, please don't hesitate to ask. As scientists, it is important for us to have a clear understanding of mathematical concepts in order to accurately interpret and analyze data. Keep up the good work in trying to understand these concepts!