compact open topology for a singleton homeomorphic to the space

1. Proposition

Let \((X, \mathcal{T})\) be a topological space. Then the mapping space \(\mathcal{C}(*,X)\) with the compact open topology for \(*\) as singleton is homeomorphic to \(X\) for

\begin{align*} \varphi: X \rightarrow& \mathcal{C}(*,X) \\ x \mapsto& (f: * \mapsto x) \end{align*}

2. Proof

Homeomorphismus als stetig abgeschlossene/offene Bijektion

2.2. bijective

follows from the evident Inverse map

\begin{align*} \varphi^{-1}: (f) \mapsto f(*) \end{align*}

2.3. continuous

continuous map and subbasis Let \(\mathcal{O}_{K,U} \subseteq \mathcal{C}(*,X)\) be open. Then it follows

\begin{align*} \varphi^{-1}[\mathcal{O}_{K,U}] =& \varphi^{-1}[\mathcal{O}_{*,U}] \\ =& \varphi^{-1}[\{f \in \mathcal{C}(*,X) \vert f(*) \in U\}] \\ =& \{x \in X \vert \varphi(x)(*) \in U \} \\ =& \{x \in X \vert x \in U \} =& U \end{align*}

2.4. open map

Suppose \(O\) is open, then

\begin{align*} \varphi[O] =& \{f \in \mathcal{C}(*,X) \vert f(*) \in O\} \\ =& \mathcal{O}_{*, O} \end{align*}