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In the MetricSpace_ZF we show how a single (ordered loop valued) pseudometric defines a uniformity. In this theory we extend this to the situation where we have an arbitrary collection of pseudometrics, all defined on the the same set \(X\) and valued in an ordered loop \(L\). Since real numbers form an ordered loop all results proven in this theory are true for the standard real-valued pseudometrics.

Suppose \(\mathcal{M}\) is a collection of (an ordered loop valued) pseudometrics on \(X\), i.e. \(d:X\times X\rightarrow L^+\) is a pseudometric for every \(d\in \mathcal{M}\). Then, for each \(d\in \mathcal{M}\) the collection of sets \(\mathfrak{B} = \{ d^{-1}(\{c\in L^+: c\leq b\}): b \in L_+ \}\) forms a fundamental system of entourages. Consequently, by taking supersets of \(\mathfrak{B}\) we can get a uniformity defined by the pseudometric (see theorem metric_gauge_base in the MetricSpace_ZF theory. Repeating this process for each \(d\in \mathcal{M}\) we obtain a collection of uniformities on \(X\). Since all uniformities on \(X\) ordered by inclusion form a complete lattice (see theorem uniformities_compl_latt in the UniformityLattice_ZF) there exists the smallest (coarsest) uniformity that contains all uniformities defined by the pseudometrics in \(\mathcal{M}\).

To shorten the main definition let's define the uniformity derived from a single pseudometric as UnifFromPseudometric. In the definition below \(X\) is the underlying set, \(L,A,r\) are the carrier, binary operation and the order relation, resp., of the ordered loop where the pseudometrics takes its values and \(d\) is the pseudometrics.

We construct the uniformity from a collection of pseudometrics \(\mathcal{M}\) by taking the supremumum (i.e. the least upper upper bound) of the collection of uniformities defined by each pseudometric from \(\mathcal{M}\).

The context muliple_pmetric is very similar to the pmetric_space context except that rather than fixing a single pseudometric \(d\) we fix a nonempty collection of pseudometrics \(\mathcal{M}\). That forces the notation for disk, topology, interior and closure to depend on the pseudometric \(d\). The \(\mathcal{U}\) symbol will denote the collection of uniformities generated by \(\mathcal{M}\). We also assume that the positive cone \(L_+\) of the loop is nonempty, \(r\) down-directs \(L_+\) (see Order_ZF for definition) and that the (positive cone of the) ordered loop is halfable (see \( MetricSpace\_ZF) \)). In short, we assume what we need for ordered loop valued pseudometrics to generate uniformities.

If \(d\) is one of the pseudometrics from \(\mathcal{M}\) then theorems proven in the pmetric_space context are valid as applied to \(d\).

In the muliple_pmetric context each element of \(\mathcal{U}\) is a uniformity.

The uniformity generated by a family of pseudometrics \(\mathcal{M}\) is indeed a uniformity which is the supremum of uniformities in \(\mathcal{U}\).

The uniformity generated by a family of pseudometrics contains all uniformities generated by the pseudometrics in \(\mathcal{M}\).

If a uniformity contains all uniformities generated by the pseudometrics in \(\mathcal{M}\) then it contains the uniformity generated by that family of pseudometrics.

The uniformity generated by the collection of pseudometrics \(\mathcal{M}\) contains all inverse images of the initial segments of the positive cone of \(L\), (i.e. sets of the form \(d^{-1}(\{c\in l: 0 \leq c \leq y \})\) for \(d\in \mathcal{M}\) and \(y\) from the positive cone \(L_+\) of \(L\)).

The uniformity generated by the collection of pseudometrics \(\mathcal{M}\) is the coarsest uniformity that contains all inverse images of the initial segments of the positive cone of \(L\) with respect to all \(d\in\mathcal{M}\).

This section presents an alternative, more direct approach to defining a uniformity generated by a collection of pseudometric. This approach is probably equivalent to the one presented in the previous section, but more complicated and hence obsolete.

The next two definitions describe the way a common fundamental system of entourages for \(\mathcal{M}\) is constructed. First we take finite subset \(M\) of \(\mathcal{M}\). Then we choose \(f:M\rightarrow L_+\). This way for each \(d\in M\) the value \(f(d)\) is a positive element of \(L\) and \(\{d^{-1}(\{c\in L^+: c\leq f(d)\}): d\in M\}\) is a finite collection of subsets of \(X\times X\). Then we take intersections of such finite collections as \(M\) varies over \(\mathcal{M}\) and \(f\) varies over all possible functions mapping \(M\) to \(L_+\). To simplify notation for this construction we split it into two steps. In the first step we define a collection of finite intersections resulting from choosing a finite set of pseudometrics \(M\), \(f:M\rightarrow L_+\) and varying the selector function \(f\) over the space of functions mapping \(M\) to the set of positive elements of \(L\).

In the second step we collect all uniform gauge sets defined above as parameter \(M\) vary over all nonempty finite subsets of \(\mathcal{M}\). This is the collection of sets that we will show forms a fundamental system of entourages.

Analogously what is done in the pmetric_space context we will write \( \text{UniformGauges}(X,L,A,r,\mathcal{M} ) \) as \( \mathfrak{B} \) in the muliple_pmetric context.

The next lemma just shows the definition of \(\mathfrak{B}\) in notation used in the muliple_pmetric.

\(\mathfrak{B}\) consists of (finite) intersections of sets of the form \(d^{-1}(\{c\in L^+:c\leq f(d)\})\) where \(f:M\rightarrow L_+\) some finite subset \(M\subseteq \mathcal{M}\). More precisely, if \(M\) is a nonempty finite subset of \(\mathcal{M}\) and \(f\) maps \(M\) to the positive set of the loop \(L\), then the set \(\bigcap_{d\in M} d^{-1}(\{c\in L^+:c\leq f(d)\}\) is in \(\mathfrak{B}\).

If \(d\) is member of any finite subset of \(\mathcal{M}\) then \(d\) maps \(X\times X\) to the nonnegative set of the ordered loop \(L\).

Members of the uniform gauge defined by multiple pseudometrics are subsets of \(X\times X\) i.e. relations on \(X\).

Suppose \(M_1\) is a subset of \(M\) and \(\mathcal{M}\). \(f_1,f\) map \(M_1\) and \(M\), resp. to \(L_+\) and \(f\leq f_1\) on \(M_1\). Then the set \(\bigcap_{d\in M} d^{-1}(\{y \in L_+: y\leq f(d)\})\) is included in the set \(\bigcap_{d\in M_1} d^{-1}(\{y \in L_+: y\leq f_1(d)\})\).

For any two sets \(B_1,B_2\) in \(\mathfrak{B}\) there exist a third one that is contained in both.

Sets in \(\mathfrak{B}\) contain the diagonal and are symmetric, hence contain a symmetric subset from \(\mathfrak{B}\).

\(\mathfrak{B}\) is a subset of the power set of \(X\times X\).

If \(\mathcal{M}\) and \(L_+\) are nonempty then \(\mathfrak{B}\) is also nonempty.

If the loop order is halfable then for every set \(B_1\in \mathfrak{B}\) there is another set \(B_2\in \mathfrak{B}\) such that \(B_2\circ B_2 \subseteq B_1\).

If \(\mathcal{M}\) is a nonempty collection of pseudometrics on a nonempty set \(X\) valued in loop \(L\) partially ordered by relation \(r\) such that the set of positive elements \(L_+\) is nonempty, \(r\) down directs \(L_+\) and \(r\) is halfable on \(L\),then \(\mathfrak{B}\) is a fundamental system of entourages in \(X\) hence its supersets form a uniformity on \(X\) and hence those supersets define a topology on \(X\)