small fix
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This commit is contained in:
Josia Pietsch 2024-01-05 17:43:25 +01:00
parent 99ca39e52c
commit 8ff3cadebd
Signed by: josia
GPG key ID: E70B571D66986A2D
2 changed files with 2 additions and 2 deletions

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@ -23,7 +23,7 @@ $X^{X}$ is a compact Hausdorff space.
\item $X^X \ni f \mapsto f \circ f_0$ \item $X^X \ni f \mapsto f \circ f_0$
is continuous: is continuous:
Consider $\{f : f f_0 \in U_{\epsilon}(x,y)\}$. Consider $\{f : f \circ f_0 \in U_{\epsilon}(x,y)\}$.
We have $ff_0 \in U_{\epsilon}(x,y)$ We have $ff_0 \in U_{\epsilon}(x,y)$
iff $f \in U_\epsilon(x,f_0(y))$. iff $f \in U_\epsilon(x,f_0(y))$.
\item Fix $x_0 \in X$. \item Fix $x_0 \in X$.

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@ -108,7 +108,7 @@ This will follow from the following lemma:
is continuous. is continuous.
Since $T_n$ is compact, Since $T_n$ is compact,
we have that $\{(x,t) \mapsto tx : t \in T_n\}$ we have that $\{(x,t) \mapsto tx : t \in T_n\}$
is equicontinuous for all $n$. is equicontinuous.
So there is $\epsilon > 0$ such that So there is $\epsilon > 0$ such that
$d(x_1,x_2) < \epsilon \implies d(tx_1, tx_2) < a -b$ $d(x_1,x_2) < \epsilon \implies d(tx_1, tx_2) < a -b$
for all $t \in T_n$. for all $t \in T_n$.