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

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 \(d\).

Proof

Pick \(x \in B_r(x_0)\). Then \[ \begin{align} d(x,x_0) < r &\quad\Longrightarrow\quad \exists\: \varepsilon > 0; \quad d(x,x_0) < r - \varepsilon\\ &\quad\Longrightarrow \quad d(y,x) < \varepsilon \quad\text{ implies }\quad d(y,x_0) \leq d(y,x) + d(x,x_0) < \varepsilon + (r - \varepsilon) = r. \end{align} \] This means: \( y \in B_r(x_0) \) if \( y \in B_\varepsilon(x)\), i.e. \( B_\varepsilon(x) \subset B_r(x_0)\).

¹⁾

An alternative to this approach is to take closed sets as complements of open sets. These two definitions, however, are completely equivalent. In particular, a set is open exactly when it does not contain its boundary.

²⁾

Equivalent norms induce the same 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.