I’d show alternate definitions of interiors, closures and boundaries for topological spaces which I’ve found immensely helpful.

Let’s go through the usual definitions first. Given a topological space $$X$$ with a subset $$S \subset X$$, the interior $$\text{int}(S)$$ of $$S$$ is defined to be the union of all open subsets of $$X$$ that is contained in $$S$$. The closure $$\overline{S}$$ is defined to be the intersection of all closed subsets of $$X$$ containing $$S$$. $$\overline{S} \backslash \text{int}(S)$$ is known as the boundary of $$S$$ and denoted $$\delta S$$.

The equivalent definitions I like to use are

\begin{align*} a \in \text{Clo}(S) \iff& \nexists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset X \backslash S,\\ a \in \text{Int}(S) \iff& \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset S,\\ a \in \delta S \iff&\nexists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset S \\ &\text{ and } \nexists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset X \backslash S. \end{align*}

The similarities in the definitions prompts us to define the following. Given some $$a \in X$$ we let $$P$$ and $$Q$$ be the statements

\begin{align*} P &: \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset S,\\ Q &: \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset X \backslash S. \end{align*}

It’s impossible for $$P$$ and $$Q$$ to both be true so we have the truth table

This means $$\text{Int}(S)$$, $$\delta S$$ and $$\text{Int}(X \backslash S)$$ partition $$X$$

## Accumulation and isolated points

Let $$S'$$ denote the set of accumulation points of $$S$$ and $$\text{Iso}(S)$$ denote the set of isolated points of $$S$$, we have the equivalent definitions

\begin{align*} a \in S' &\iff \nexists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \backslash \{a\} \subset X \backslash S, \\ a \in \text{Iso}(S) &\iff \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \backslash \{a\} \subset X \backslash S\text{ and } a \in S. \\ \end{align*}

This shows that we’re concerned with $$U \backslash \{a\}$$ for open sets $$U$$ that contain $$a$$. Inspired by how we defined $$P$$ and $$Q$$, we define the following statements $$V$$, $$W$$ and $$K$$ for some $$a \in X$$ as follows.

\begin{align*} P &: \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset S,\\ Q &: \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \subset X \backslash S, \\ V&: \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \backslash \{a\} \subset S, \\ W&: \exists \text{ open } U \subset X \text{ s.t. } a \in U \text{ and } U \backslash \{a\} \subset X \backslash S, \\ K&: a \in S. \end{align*}

Using this language, we can see the following equivalences.

\begin{align*} a \in S' &\iff \neg W, \\ a \in \text{Iso}(S) &\iff W \land K. \end{align*}

Also note that

\begin{align*} P &\iff V \land K, \\ Q &\iff W \land \neg K. \end{align*}

As such $$\neg Q \iff \neg W \lor K \iff (\neg W) \text{ XOR } (W \land K)$$. So accumulation points and isolated points form a partition of the closure.

We can further split into cases. Considering the interior of $$S$$ first, notice that $$P \land \neg Q \iff V \land K$$. So we have the following cases

Considering the boundary $$\delta S$$, we also have $$\neg P \land \neg Q \iff (K \land \neg V) \lor (\neg K \land \neg W)$$. So we have

We can conduct a similar analysis for the interior of $$X \backslash S$$.

## Examples

Consider the real line with the topology induced from the Euclidean metric. Let $$S=((0,2] \cup \{3\})\backslash\{1\}$$. Then we have

Note that $$\text{Int}(S) \cap \text{Iso}(S) = \emptyset$$ for all subsets $$S$$ of $$\mathbb{R}$$. However if we consider some discrete topological space $$X$$, then $$\text{Int}(S) \cap \text{Iso}(S) = S$$ for all subsets $$S$$ of $$X$$.