Definition of disconnected?

Question

I am confused in definition of disconnected set in topological space $X$.

Which of the following definition is valid/correct?

1) a subset $A$ of $X$ is disconnected if $A=V ∪W$ where $U$ and $V$ are non-empty disjoint open sets in $X$

2)a subset $A$ of $X$ is disconnected if $A⊆ V ∪W$ where $U$ and $V$ are non-empty disjoint open sets in $X$

Further I saw, an answer on MSE that, $\mathbb{Q}×[0,1]$ is not connected subspace in $\mathbb{R^2}$ with usual topology! the answer is like this!!

" As for any $r∈\mathbb{R}-\mathbb{Q}$ we can divide, given subspace into two nonempty disjoint open sets $(-∞,r)×[0,1]$ and $(r,∞)×[0,1]$ hence $\mathbb{Q}×[0,1]$ is not connected in $\mathbb{R^2}$ "

What they mean by "divide"? Please help me....

Answer 1

The first definition is correct. The second one cannot be correct. We see this by letting $A = V$. Then $A$ would be a connected component of a disconnected space, but $A$ itself is still connected. When considering connected spaces, we want to capture the notion of some "divide" or "void" between connected components. In the language of open sets, we phrase this as being able to "divide" the space into a union of disjoint open sets. In other words, "divide" just means that you are able to find such a union between disjoint open sets.

Answer 2

Let $T_X$ be the topology on $X.$ The subspace topology $T_A$ on $A\subset X$ is $T_A=\{t\cap A:t\in T\}.$

Any space is disconnected iff it is the union of two disjoint non-empty open sets.

So the sub$space$ $A$ is disconnected iff $A=B\cup C$ where $B,C$ are disjoint non-empty members of $T_A.$

Equivalently, the subspace $A$ is disconnected iff there exist $U,V\in T_X$ such that $U\cap V\supset X$ and such that $U\cap A, V\cap A$ are non-empty and disjoint.

Neither def'n 1) nor def'n 2) is correct.