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\title{Fun theorems}
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\begin{itemize}
\item In Fr\'{e}chet (T\tS{1}) spaces, points are closed, and every subset is the intersection of all open sets containing that subset.
\item All spaces are the quotient of some Hausdorff space.
\item Compact sets are closed in Hausdorff spaces.
\item If $f, g: X \rightarrow Y$ are continuous functions and $Y$ is Hausdorff, then their equaliser is closed in $X$.
\item Continuous functions into Hausdorff spaces are determined by their values on dense subsets.
\item Completely regular spaces have their topology determined by their set of continuous functions to $\R$.
\item Tychonoff spaces are exactly those spaces which can be embedded in compact Hausdorff spaces.
\item Every space $X$ is a subspace of some separable space with the same cardinality as $X$.
\item Continuous functions from separable spaces to Hausdorff spaces are determined by countably many values.
\item Separable metric spaces can be embedded in the Hilbert cube.
\item $X$ is sequential iff for all spaces $Y$ and functions $f: X \rightarrow Y$, $f$ is continuous iff for all sequences $(x_n)$ and points $x$, $x_n \rightarrow x \Rightarrow f(x_n) \rightarrow f(x)$.
\item The topology of second-countable spaces has cardinality less than or equal to that of the real line.
\item Continuous images of compact spaces are compact.
\item Any product of compact spaces is compact.
\item Continuous images of (\{\}/path-/hyper-) connected spaces are (\{\}/path-/hyper-) connected.
\item Every Hausdorff second-countable regular space is metrisable.
\item For metrisable spaces, second-countability, separability, and Lindel\"{o}fness are equivalent, as are compactness, sequential compactness, countable compactness, and limit point compactness.
\item For first-countable spaces, sequential compactness and countable compactness are equivalent.
\item For Lindel\"{o}f spaces, compactness and countable compactness are equivalent.
\end{itemize}
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