Connected Sets in Topology

Question

Theorem: Let $(X,\mathscr{T})$ be a topological space. If $E$ is connected and $K$ is such that $E\subseteq K\subseteq\mathrm{cl}(E)$, then $K$ is connected. (Cl(E) is closure of E)

Question: Consider the standard topology on $\mathbf{R}$. Let $\mathbf{E}$ = (2,4). Then cl($\mathbf{E}$)= [2,4]. Let $\mathbf{B}$ = [2,4) My claim is that B is disconnected since I can partition it with [2,3) and [3,4).

What am I doing wrong?

(I have just started studying topology on my own so it is very possible that mine is a stupid mistake.)

Answer 1

The point is that $[3,4)$ is not open set in $[2,4)$. In the definition of connectivity it is needed that there exist no two open sets such that $X = A\cup B$ and if there exist such sets $X$ is not connected.

Answer 2

A set is only disconnected when we can partition it into two disjoint non-empty open sets, not just two sets. And $[2,3)$ is not open in the standard topology on $\mathbb{R}$. Neither is $[3,4)$.

Why don't you partition $(2,4)$ in $(2,3)$ and $[3,4)$, e.g.? The same would hold, and still you believe/know that $(2,4)$ is connected.