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Using the standard topology on $\mathbb{R}^d$, interior of a subset $S$ of topological space $M$ is defined as the set of all elements of $S$ except the boundary of $S$.

However, I only understand the boundary in this sense to be defined with the standard topology on $\mathbb{R}^d$, so how would we define boundary and "interior" in topological spaces in general?

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    https://en.wikipedia.org/wiki/Interior_(topology) – 2017-01-31
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    Of course it is. The topological definition if the interior is : Interior of $X$ is the biggest open $O$ s.t. X\supset O$ – 2017-01-31
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    @Surb. Ah of course. And then the boundary is the intersection of the complement of $O$ with $X$? – 2017-01-31
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    And in case you wonder, the definition Surb gives makes sense: Since the empty set is open and $\emptyset \subset X$, there is an open contained in $X$, and since the union of opens is again open, we can define the interior as the union of all opens contained in $X$, which clearly is the biggest open contained in $X$. – 2017-01-31

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Let $(X,\mathcal{T})$ be a topological space. For $A \subseteq X$ we define the interior of A as $$\operatorname{Int}A:=\bigcup\{C \subseteq X : C \subseteq A, C \in \mathcal{T}\}$$ Furthermore, the exterior of A is $$\operatorname{Ext}A := X \setminus \overline{A}$$ where $\overline{A}$ is the closure of A defined by $$\overline{A}:=\bigcap\{B \subseteq X : B \supseteq A, B^c \in \mathcal{T}\}$$ Then we define the boundary of A as $$\partial A := X \setminus (\operatorname{Int}A \cup \operatorname{Ext}A)$$ Sure, this notions are very abstract. But they reduce to the familiar in for example metric spaces like $\mathbb{R}^n$. Howevery, many properties can be in fact proved in this general setting. Like for example if a sequence $x_n$ in $A$ converges to $x$, then $x \in \overline{A}$.

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    What is $B^c$ in the closure definition? The complement of $B$? – 2017-01-31
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    @Programmer2134 Yes, $B^c$ denotes the complement of $B$ in $X$. In words: $B^c \in \mathcal{T}$ just means that $B$ is closed in $X$. – 2017-01-31
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By definition, the interior of a set $S$ is defined as the largest open set contained in $S$.

For this definition to "work", all you need is a topology, there is no need for it to be the standard topology on $\mathbb R^d$.

Equivalently, you can define the interior of $S$ (denoted $\mathrm{int}(S)$)as:

  1. The largest open set (by the relation of inclusion) contained in $S$. This means that $\mathrm{int}(S)$ is defined by
    • $\mathrm{int}(S)\subseteq S$
    • $\mathrm{int}(S)$ is open
    • if $O$ is open and $O\subseteq S$, then $O\subseteq \mathrm{int}(S)$.
  2. The union of all open sets contained in $S$, i.e. $$\mathrm{int}(S)=\bigcup_{O\text{ is open}\\ O\subseteq S} O$$
  3. The set of all interior points, i.e. $$\mathrm{int}(S)=\{x\in S: \exists \text{ open set }O: O\subseteq S \land x\in O\}$$