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Question: If we start $C[0,1]$ which we let be the space of continuous functions on $[0,1]$ equipped with the metric $$ d(f,g)=\sup_{x\in [0,1]} |f(x)-g(x)| $$ and I have some set $$ H=\{h:[0,1]\rightarrow \mathbb{R}\}$$ Now I want to to try and find the interior, boundary and closure of $H$ in $C[0,1]$.

I'm confused about how one might even approach a problem such as this.

Thank you for any help.

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  • $\begingroup$ What do you mean by standard metric? The supremum metric is the standard metric of $C[0,1]$... as for your question: can you think of a sequence $f_n\notin H$ with $f_n\to 0$? Also, if $g_n\in H$ converges to $g$, what can you say about $g_n(1)$? $\endgroup$ – user251257 Mar 26 '17 at 15:29
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$H$ is limit point closed and therefore closed.

Let $G$ be the complement of $H$ in $\mathcal{C}[0, 1]$. $G$ can approximate any function in $H$ arbitrarily well (for example, if $h(x) \in H$, then $h_n(x) := h(x) + \frac{1}{n}$ has $\lim_{n \to \infty} h_n(x) = h(x)$). Therefore \begin{align*} Int(H)^{c} = \overline{G} = \mathcal{C}[0, 1] \end{align*} so that Int$(H) = \emptyset$.

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A less topological answer: $H$ is a $1$ dimensional subspace, and thus, is closed. As $\dim C[0,1]>1$, it can't be open either.

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