Intersection of Connected Sets

Re-corrected: $\newcommand{\cl}{\operatorname{cl}}$Suppose that $Y=H\cup K$, where $H\cap K=\varnothing$, $H\ne\varnothing\ne K$, and $H$ and $K$ are clopen in $Y$. Let $f:Y\to\{0,1\}$ take $H$ to $0$ and $K$ to $1$, use the Tietze extension theorem to extend $f$ continuously to $\hat f:X\to[0,1]$, and let

$$\begin{align*} U&=\hat f^{-1}\left[\left[0,\frac12\right)\right]\supseteq H\;,\\ V&=\hat f^{-1}\left[\left(\frac12,1\right]\right]\supseteq K\;,\text{ and}\\ F&=\hat f^{-1}\left[\left\{\frac12\right\}\right]\;. \end{align*}$$

For each $A\in\phi$ we must have $F\cap A\ne\varnothing$, as otherwise $U\cap A$ and $V\cap A$ would be a disconnection of $A$. Thus, $\mathscr{C}=\{F\cap A:A\in\phi\}$ is a nested family of non-empty compact sets. But then $\varnothing\ne\bigcap\mathscr{C}=F\cap Y=\varnothing$, which is absurd.