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

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

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