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\markright{Point Set Topology:  Kai Tsuruta}
\author{Kai Tsuruta}


\title{Point Set Topology Table}

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%\section{The Continuous Image of a Lindelof Space is also Lindelof}



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{\bf The continuous image of a Lindel$\ddot{o}$f space is also Lindel$\ddot{o}$f
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Let $f: X \rightarrow f(X)$ be a continuous map and $\mathcal{A}$ be any covering of $f(X)$. Then $\left\{f^{-1}(A) | A \in \mathcal{A}\right\}$ is a collection of sets covering $X$; these sets are open in $X$ by the continuity of $f$. Then, since $X$ is a Lindel$\ddot{o}f$ space, $\exists$ a countable subcover  $\left\{f^{-1}(A_n)\right\}$ of $\left\{f^{-1}(A) | A \in \mathcal{A}\right\}$. Then $\left\{A_n  \right\}$ is a countable subcover of the original cover $\mathcal{A}$. Thus, $f(X)$ is Lindel$\ddot{o}$f.



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