# Thread: Proving the compactness and connectedness of a subset

1. ## Proving the compactness and connectedness of a subset

Any hint on how to approach this problem?

Thanks as always

2. Originally Posted by 6DOM

Any hint on how to approach this problem?

Thanks as always
Let $g:[a,b]\longrightarrow\Gamma_f$ be $g(x)=(x,f(x))$. Let $\{C_i\}$ be an open cover of $\Gamma_f$. The continuity of $g$ implies that $g^{-1}(C_i)$ is open in $[a,b]$ (this is a theorem) and $\{g^{-1}(C_i)\}$ is an open cover of $[a,b]$.

Spoiler:
Since $[a,b]$ is compact, we know that there are finitely many $C_i$ such that $[a,b]\subset\bigcup_{i=1}^N g^{-1}(C_i)$. Because $g(g^{-1}(C_i))\subseteq C_i$, we know that $\Gamma_f\subset\bigcup_{i=1}^N C_i$. So $\Gamma_f$ is compact. $\square$

You want to approach connectedness similarly. Assume that $\Gamma_f\subset U\cup V$ where $U$ and $V$ are open sets such that $cl(U)\cap V=U\cap cl(V)=\emptyset$ and try to derive a contradiction. (Your contradiction will be that $[a,b]$ is not connected.)