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Let $K$ be a Hausdorff compact set and take a net $(x_\lambda)$ and a point $x$ in $K$. Suppose that $f(x_\lambda)$ converges to $f(x)$ for all $f \in C(K, \mathbb R)$. How can one proof that $x_\lambda \to x$?

In this problem, I'm considering real-valued functions $K \to \mathbb R$.

I thought in approuch this problem in the following way:

Let $\delta_y : C(K) \to \mathbb R$ be the the functional such that $\delta_y(f) = f(y)$. Then, we have that $\delta_{x_\lambda}(f) \to \delta_x (f)$ for all $f \in C(K)$ and hence $\delta_{x_\lambda} \to \delta_x$ in the weak-star topology of $[C(K)]^{'}$. However I don't know how to get that $x_\lambda \to x$.

Help?

user 242964
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1 Answers1

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If not, then there is a subnet $\{x_{\lambda_i}\}_{i\in J}$ and an open set $U\subseteq K$ such that $\{x_{\lambda_i}\}_{i\in J}\subseteq \overline U$ and $\overline U\cap \{x\}=\emptyset.$ Urysohn provides a continuous function $f:K\to [0,1]$ such that $f(\overline U)=0$ and $f(\{x\})=1.$ And since $\{x_{\lambda_i}\}_{i\in J}\subseteq \overline U$, $f(x_{\lambda})\nrightarrow f(x).$

Matematleta
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