\[ \newcommand{R}{\mathbb{R}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} \]

# Interior points, boundary points, open and closed sets

Let \((X,d)\) be a metric space with distance \(d\colon X \times X \to [0,\infty)\).

- A point \(x_0 \in D \subset X\) is called an
**interior point in D**if there is a small ball centered at \(x_0\) that lies entirely in \(D\),

\[ x_0 \text{ interior point } \defarrow \exists\: \varepsilon > 0; \qquad B_\varepsilon(x_0) \subset D. \]

- A point \(x_0 \in X\) is called a
**boundary point of D**if any small ball centered at \(x_0\) has non-empty intersections with both \(D\) and its complement,

\[ x_0 \text{ boundary point } \defarrow \forall\: \varepsilon > 0 \quad \exists\: x,y \in B_\varepsilon(x_0); \quad x \in D,\: y \in X \setminus D. \]

- The set of interior points in D constitutes its
**interior**, \(\mathrm{int}(D)\), and the set of boundary points its**boundary**, \(\partial D\). \(D\) is said to be**open**if any point in \(D\) is an interior point and it is**closed**if its boundary \(\partial D\) is contained in \(D\); the**closure of D**is the union of \(D\) and its boundary:

\[
\overline D := D \cup \partial D.
\]
\(\qquad \)Alternative notations for the closue of \(D\) in \(X\) include \(\overline{{D\,}^X}\), \(\mathrm{clos}(D)\) and \(\mathrm{clos}(D;X)\).^{1)}

**Ex.**

- In \(\R\) with the usual distance \(d(x,y) = |x-y|\), the interval \((0,1)\) is open, \( [0,1) \) neither open nor closed, and \( [0,1] \) closed.
^{2)} - The set \[D := \{(x,y) \in \R^2 \colon x > 0, y \geq 0\}\] is neither closed nor open in Euclidean space \(\R^2\) (metric coming from a norm, e.g., \(d(x,y) = \|x-y\|_{l_2} = ((x_1-y_1)^2 + (x_2-y_2)^2)^{1/2}\)), since its boundary contains both points \((x,0)\), \(x > 0\), in \(D\) and points \((0,y)\), \(y \geq 0\), not in \(D\). The closure of D is

\[ \overline D = \{(x,y) \in \R^2 \colon x \geq 0, y \geq 0\}. \]

- An entire metric space is both open and closed (its boundary is empty).
- In \(l_\infty\), \[ B_1 \not\ni (1/2,2/3,3/4,\ldots) \in \overline{B_1}.\]
- For a general metric space, the
**closed ball**\[\tilde B_r(x_0) := \{ x \in X\colon d(x,x_0) \leq r\}\] may be larger than the closure of a ball, \(\overline{B_r(x_0)}\). If we let \(X\) be a space with the discrete metric, \[ \begin{cases} d(x,x) &= 0,\\ d(x,y) &= 1, \quad x\neq y. \end{cases} \] Then \[ B_1(x_0) = \{x_0\}, \quad\text{ so that }\quad \overline{B_1(x_0)} = \overline{\{x_0\}} = \{x_0\}. \] But \[ \tilde B_1(x_0) = X.\]

### ℘ (Open) balls are open

Let \((X,d)\) be a metric space, \( x_0\) a point in \(X\), and \(r > 0\). Then \(B_r(x_0)\) is open in \(X\) with respect to the metric \(d\).

^{1)}

^{2)}

**topology**on a space (i.e., the same open and closed sets). Since all norms on \(\R^n\) are equivalent, it is unimportant which norm we choose.