Weronika

The part (a) is not answered here - so let me include it, for the sake of completeness.$\newcommand{\inv}[1]{{#1}^{-1}}$ We want to show that $\inv{(g\circ f)}(Z)=\inv f(\inv g(Z))$. For any $w\in W$ we have: $w\in \inv{(g\circ f)}(Z)$ $\Leftrightarrow$ $(g\circ f)(w)\in Z$ $\Leftrightarrow$ $g(...

Let $~\mathcal{U}, \mathcal{U}'$ be bases for $V$ and $W$, respectively. Suppose $f : V \rightarrow W$ is continuous. Let $v_0 \in V$ and $ U \in \mathcal{U}'$ such that $f(v_0) \in U$. Show that there exists $T \in \mathcal{U}$ such that $f(T) \subseteq U $. My approach: * still editing * By def...