Exercise 2.5.2 (Continuous identity to cofinite topology)

Question

Let $X$ be a topological space and let $X_0$ be the topological space that is the set $X$ with the cofinite topology. Show that the identity map of $X$ to $X_0$ is continuous if and only if $X$ is a $T_1$-space.

Elu Thingol · Accepted Answer

\begin{proof}
\begin{description}
\item[$\implies$] Suppose that $X$ is a $T_1$-space, i.e., all singletons $\{x\}$, $x \in X$, are closed. Then all finite sets must be closed as well. Thus, the cofinite sets are open. As a consequence, $I:X 	o X_0$ must be open, since the inverse of every cofinite set in $X_0$ is open in $X$.

\item[$\impliedby$] Suppose that the identity map $I: X 	o X_0$ is continuous. By \href{./exercise_2-3-2}{Exercise 2.3.2}, the inverse image of every closed set under $I$ is closed; in particular, singleton sets $I^{-1}(\{x\}) = \{x\}$, $x \in X$, are closed, and therefore $X$ is a  $T_1$-space.
\end{description}
\end{proof}

Exercise 2.5.2 (Continuous identity to cofinite topology)

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Exercise 2.5.2 (Continuous identity to cofinite topology)

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