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Let $K$ be a compact Hausdorff space such that the set $D$ of isolated points in $K$ is countable and dense in $K$. Consider the linear subspace $A$ of $C(K)$ consisting of those functions $f\in C(K)$ such that $f$ is constant on $K\setminus D$ and the set $\{x\in D\colon f(x)= 0\}$ is finite or its complement in D is finite.

Is $A$ a closed subspace of $C(K)$?

  • 1
    Is this homework?2011-11-23
  • 0
    first, do you have an example of compact space whith dense isolated points that has non-isolated points ?2011-11-23
  • 0
    What is the topology of $C(K)$?2011-11-23
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    $C(K)$ is regarded as a Banach space with the supremum norm.2011-11-23
  • 0
    So what a neighborhood of a function $f \in C(K)$ would be like?2011-11-23
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    Typically, a ball $\{g\in C(K): \sup_{x\in K}|f(x)-g(x)|<\varepsilon\}$.2011-11-23

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