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The Open Mapping TheoremWe are finally going to prove the open mapping theorem in $F$-space. In this version, only metric and completeness are required. Therefore it contains the Banach space version naturally.(Theorem 0) Suppose we have the following conditions:$X$ is a $F$-space,$Y$ is a topological space,$\Lambda: X \to Y$ is continuous and linear, and$\Lambda(X)$ is of the second category in $Y$.Then $\Lambda$ is an open mapping.Proof. Let $B$ be a neighborhood of $0$ in $X$. Let $d$ be an i...