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Let $(M,d)$ be a metric space. Then $ x \in S $ is an interior point of $ S $ if some ball centered around S of positive radius is wholly contained in $S$.

But consider this. The set $S_{L}$ of all strings of length $L$ is a metric space under the Hamming distance. Let $S \subset S_{L}$ and Let $s \in S$ be any string in $S$. Then technically, the ball of radius 0.5 centered on $s$ must be be wholly contained in $S$ since this ball can contain no string other than $s$ itself. Therefore, $s$ is an interior point of $S$, therefore $S$ is open. Does this mean that every subset of $S_{L}$ is necessarily open, or is there some major flaw that I have missed in the definition of an interior point(which I am quite sure I have)?

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  • $\begingroup$ It is quite possible to have sets $S$ with empty interior, or that the interior of $S$ could be all of $S$. (The latter means that $S$ is an open set, but the former does not imply that $S$ is empty.) $\endgroup$
    – hardmath
    Jan 27, 2016 at 16:03

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Yes, every subset is open (and closed). We get what is called the discrete topology.

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