Definition 2.2.1. Let \(P\) and \(R\) be a point sort and a scalar sort, respectively. Then, we let
\[\mathcal L_{\mathrm{met}} = \left\{ P,R,\, 0_R,+_R,<_{R},\, d \right\},\]where \(0_R\) is a constant of sort \(R, +_R:R^2\to R, <_{R}\subseteq R^2,\) and \(d:P^2\to R\) is a function symbol. An \(\mathcal L_{\mathrm{met}}\)-structure therefore has the form
\[\mathcal M = \left( M,S;\, 0^{\mathcal M},+^{\mathcal M},<^{\mathcal M},d^{\mathcal M} \right),\]where \(M\) is the set of points and \(S\) is the set in which distances take their values.
Suppose that the scalar sort is interpreted as the ordered additive structure of the real numbers. That is,
\(\mathcal M_{(M,d)}\) is a metric space if it satisfies the following sentences, where \(x,y,z\) range over the point sort \(P\!:\)
- \(\forall x,y\ (0\le d(x,y)),\)
- \(\forall x,y\ (d(x,y)=0\leftrightarrow x=y),\)
- \(\forall x,y\ (d(x,y)=d(y,x)),\)
- \(\forall x,y,z\ \left(d(x,z)\le d(x,y)+d(y,z)\right).\)
In this case, \(d\) is called the metric on \(M.\)
Definition 2.2.2. A set \(S\subseteq \mathbb R^k\) is convex if
\[\forall x,y\in S\ \forall \lambda \in (0,1)\ \left( \lambda x + (1-\lambda)y\in S\right).\]Example 2.2.3. Let \(a_i<b_i\) for \(i=1,\dots, k,\) and define \(S\) to be the set of all points \(x=(x_1,\dots,x_k)\) in \(\mathbb R^k\) such that \(x_i\in [a_i,b_i]\) for \(i=1,\dots, k.\) The set \(S\) is called a \(k\)-cell.
Let \(x,y\in S, \lambda \in (0,1),\) and \(z_i = \lambda x_i + (1-\lambda)y_i.\) Since
we have \(a_i\le z_i\le b_i,\) so that \(z_i\in S.\) Thus \(S\) is convex.
In general, \(k\)-cells are convex.
Definition 2.2.4. Let \((M,d)\) be a metric space. For \(a\in M\) and \(r\in\mathbb R_{>0}\), the open ball of radius \(r\) centered at \(a\) is defined externally by
\[B_r(a)=\{x\in M:d(x,a)<r\}.\]The corresponding closed ball and punctured open ball are
\[\overline B_r(a)=\{x\in M:d(x,a)\le r\}\]and
\[B_r^{\ast}(a)=B_r(a)\setminus\{a\}=\{x\in M:0<d(x,a)<r\}.\]Example 2.2.5. Let \((M,d)\) be a metric space where
\[d(x,y)= \begin{cases} 0,&x=y,\\ 1,&x\ne y. \end{cases}\]For \(a\in M,\) if \(0<r\le 1\) then \(B_r(a)=\lbrace a\rbrace,\) and if \(1<r\) then \(B_r(a) = M.\)
Definition 2.2.6. Let \((M,d)\) be a metric space. A subset \(N\subseteq M\) is a neighborhood of \(a\in M\) if
\[\exists r>0\ \left( B_r(a)\subseteq N\right) .\]Definition 2.2.7. Let \((M,d)\) be a metric space, and let \(A\subseteq M.\) The interior of \(A\) is the set \(\operatorname {int} A\) defined by
\[\operatorname{int} A = \lbrace x\in M : \exists r>0 \ \left(B_r(x)\subseteq A\right)\rbrace .\]\(a\in A\) is an interior point of \(A\) if \(a\in \operatorname {int} A.\)
The closure of \(A\) is the set \(\overline A\) defined by
\(a\) is a closure point of \(A\) if \(a\in \overline A.\)
Let
\(a\) is a limit point of \(A\) if \(a\in A'.\) If \(a\in A\) and \(a\notin A'\) then \(a\) is an isolated point of \(A.\)
Definition 2.2.8. Let \((M,d)\) be a metric space, and let \(A\subseteq M.\)
The set \(A\) is open if
\(A\) is closed if
\[A=\overline A,\]\(A\) is perfect if
\[A=A',\]and \(A\) is dense in \(M\) if
\[\overline A=M.\]The set \(A\) is bounded if
\[\exists x\in M\ \exists r>0\ (A\subseteq B_r(x)).\]Theorem 2.2.9. Let \((M,d)\) be a metric space, and let \(A\subseteq M.\) Then, \(A\) is open if and only if \(A\) is a neighborhood of every point of \(A.\)
References
- Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). McGraw-Hill Education.