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 the 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 sets \(\{ d^{-1}(\{c\in L^+: c\leq b\}): b \in L_+ \}\) form a fundamental system of entourages (see MetricSpace_ZF).
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 function 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 \(L\).
definition
\( \text{UniformGaugeSets}(X,L,A,r,M) \equiv \) \( \{(\bigcap d\in M.\ d^{-1}(\{c\in \text{Nonnegative}(L,A,r).\ \langle c,f(d)\rangle \in r\})).\ f\in M\rightarrow \text{PositiveSet}(L,A,r)\} \)
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.
definition
\( \text{UniformGauges}(X,L,A,r,\mathcal{M} ) \equiv \bigcup M\in \text{FinPow}(\mathcal{M} )\setminus \{\emptyset \}.\ \text{UniformGaugeSets}(X,L,A,r,M) \)
The context muliple_pmetric is very similar to the pmetric_space context except that rather than fixing a single pseudometric \(d\) we fix a collection of pseudometrics \(\mathcal{M}\). That forces the notation for disk, topology, interior and closure to depend on the pseudometric \(d\).
locale muliple_pmetric = loop1 +
assumes mpmetricAssm: \( \forall d\in \mathcal{M} .\ \text{IsApseudoMetric}(d,X,L,A,r) \)
defines \( disk(d,c,R) \equiv \text{Disk}(X,d,r,c,R) \)
defines \( \tau (d) \equiv \text{MetricTopology}(X,L,A,r,d) \)
defines \( int(d,D) \equiv \text{Interior}(D,\tau (d)) \)
defines \( cl(d,D) \equiv \text{Closure}(D,\tau (d)) \)
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.
abbreviation (in muliple_pmetric)
\( \mathfrak{B} \equiv \text{UniformGauges}(X,L,A,r,\mathcal{M} ) \)
The next lemma just shows the definition of \(\mathfrak{B}\) in notation used in the muliple_pmetric.
lemma (in muliple_pmetric) mgauge_def_alt:
shows \( \mathfrak{B} = (\bigcup M\in \text{FinPow}(\mathcal{M} )\setminus \{\emptyset \}.\ \{(\bigcap d\in M.\ d^{-1}(\{c\in L^+.\ c\leq f(d)\})).\ f\in M\rightarrow L_+\}) \) unfolding UniformGaugeSets_def, UniformGauges_def\(\mathfrak{B}\) 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 \(\mathfrak{B}\).
lemma (in muliple_pmetric) mgauge_finset_fun:
assumes \( M\in \text{FinPow}(\mathcal{M} ) \), \( M\neq \emptyset \), \( f:M\rightarrow L_+ \)
shows \( (\bigcap d\in M.\ d^{-1}(\{c\in L^+.\ c\leq f(d)\})) \in \mathfrak{B} \) using assms, mgauge_def_altIf \(d\) is one of the pseudometrics from \(\mathcal{M}\) then theorems proven in pmetric_space can are valid.
lemma (in muliple_pmetric) pmetric_space_valid_in_mpm:
assumes \( d\in \mathcal{M} \)
shows \( pmetric\_space(L,A,r,d,X) \)proofIf \(d\) is member of any finite subset of \(\mathcal{M}\) then \(d\) maps \(X\times X\) to the nonnegative set of the ordered loop \(L\).
lemma (in muliple_pmetric) each_pmetric_map:
assumes \( M\in \text{FinPow}(\mathcal{M} ) \) and \( d\in M \)
shows \( d: X\times X \rightarrow L^+ \) using assms, pmetric_space_valid_in_mpm, pmetric_properties(1) unfolding FinPow_defMembers of the uniform gauge defined by multiple pseudometrics are subsets of \(X\times X\) i.e. relations on \(X\).
lemma (in muliple_pmetric) muniform_gauge_relations:
assumes \( B\in \mathfrak{B} \)
shows \( B\subseteq X\times X \)proofSuppose \(M_1\) 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 \(\bigcap_{d\in M_1} d^{-1}(\{y \in L_+: y\leq f_1(d)\})\).
lemma (in muliple_pmetric) fun_inter_vimage_mono:
assumes \( M_1\subseteq \mathcal{M} \), \( M_1\subseteq M \), \( M_1\neq \emptyset \), \( f_1:M_1\rightarrow L_+ \), \( f:M\rightarrow L_+ \) and \( \forall d\in M_1.\ f(d)\leq f_1(d) \)
shows \( (\bigcap d\in M.\ d^{-1}(\{c\in L^+.\ c\leq f(d)\})) \subseteq (\bigcap d\in M_1.\ d^{-1}(\{c\in L^+.\ c\leq f_1(d)\})) \)proofFor any two sets \(B_1,B_2\) in \(\mathfrak{B}\) there exist a third one that is contained in both.
lemma (in muliple_pmetric) mgauge_1st_cond:
assumes \( r \text{ down-directs } L_+ \), \( B_1\in \mathfrak{B} \), \( B_2\in \mathfrak{B} \)
shows \( \exists B\in \mathfrak{B} .\ B\subseteq B_1\cap B_2 \)proofSets in \(\mathfrak{B}\) contain the diagonal and are symmetric, hence contain a symmetric subset from \(\mathfrak{B}\).
lemma (in muliple_pmetric) mgauge_2nd_and_3rd_cond:
assumes \( B\in \mathfrak{B} \)
shows \( id(X)\subseteq B \), \( B = converse(B) \), \( \exists B_2\in \mathfrak{B} .\ B_2 \subseteq converse(B) \)proof\(\mathfrak{B}\) is a subset of the power set of \(\)X\times X$.
lemma (in muliple_pmetric) mgauge_5thCond:
shows \( \mathfrak{B} \subseteq Pow(X\times X) \) using muniform_gauge_relationsIf \(\mathcal{M}\) and \(L_+\) are nonempty then \(\mathfrak{B}\) is also nonempty.
lemma (in muliple_pmetric) mgauge_6thCond:
assumes \( \mathcal{M} \neq \emptyset \) and \( L_+\neq \emptyset \)
shows \( \mathfrak{B} \neq \emptyset \)proofIf 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 \(B_2\circ B_2 \subseteq B_1\).
lemma (in muliple_pmetric) mgauge_4thCond:
assumes \( \text{IsHalfable}(L,A,r) \), \( B_1\in \mathfrak{B} \)
shows \( \exists B_2\in \mathfrak{B} .\ B_2\circ B_2 \subseteq B_1 \)proofassumes \( d\in \mathcal{M} \)
shows \( pmetric\_space(L,A,r,d,X) \)assumes \( M\in \text{FinPow}(\mathcal{M} ) \) and \( d\in M \)
shows \( d: X\times X \rightarrow L^+ \)assumes \( f:X\rightarrow Y \)
shows \( f^{-1}(A) = \{x\in X.\ f(x) \in A\} \)assumes \( \forall i\in I.\ P(i) \subseteq X \)
shows \( (\bigcap i\in I.\ P(i)) \subseteq X \)assumes \( I\subseteq M \), \( I\neq \emptyset \)
shows \( (\bigcap i\in M.\ P(i)) \subseteq (\bigcap i\in I.\ P(i)) \)assumes \( b_1\leq b_2 \)
shows \( d^{-1}(\{c\in L^+.\ c\leq b_1\}) \subseteq d^{-1}(\{c\in L^+.\ c\leq b_2\}) \)assumes \( r \text{ down-directs } L_+ \)
shows \( \text{IsDownDirectedSet}(L_+,r) \)assumes \( \text{IsDownDirectedSet}(Y,r) \), \( A\in \text{FinPow}(X) \), \( B\in \text{FinPow}(X) \), \( f:A\rightarrow Y \), \( g:B\rightarrow Y \)
shows \( \exists h\in A\cup B\rightarrow Y.\ (\forall x\in A.\ \langle h(x),f(x)\rangle \in r) \wedge (\forall x\in B.\ \langle h(x),g(x)\rangle \in r) \)assumes \( A \in \text{FinPow}(X) \) and \( B \in \text{FinPow}(X) \)
shows \( A \cup B \in \text{FinPow}(X) \)assumes \( M_1\subseteq \mathcal{M} \), \( M_1\subseteq M \), \( M_1\neq \emptyset \), \( f_1:M_1\rightarrow L_+ \), \( f:M\rightarrow L_+ \) and \( \forall d\in M_1.\ f(d)\leq f_1(d) \)
shows \( (\bigcap d\in M.\ d^{-1}(\{c\in L^+.\ c\leq f(d)\})) \subseteq (\bigcap d\in M_1.\ d^{-1}(\{c\in L^+.\ c\leq f_1(d)\})) \)assumes \( x\in X \) and \( \phi (x) \)
shows \( \exists x\in X.\ \phi (x) \)assumes \( B\in \mathfrak{B} \)
shows \( id(X)\subseteq B \)assumes \( B\in \mathfrak{B} \)
shows \( B = converse(B) \)assumes \( B\in \mathfrak{B} \)
shows \( B\subseteq X\times X \)assumes \( X\neq \emptyset \)
shows \( \text{FinPow}(X)\setminus \{\emptyset \} \neq \emptyset \)assumes \( c\in Y \)
shows \( \text{ConstantFunction}(X,c) : X\rightarrow Y \)assumes \( M\in \text{FinPow}(\mathcal{M} ) \), \( M\neq \emptyset \), \( f:M\rightarrow L_+ \)
shows \( (\bigcap d\in M.\ d^{-1}(\{c\in L^+.\ c\leq f(d)\})) \in \mathfrak{B} \)assumes \( Finite(X) \), \( \forall x\in X.\ \exists y\in Y.\ P(x,y) \)
shows \( \exists f\in X\rightarrow Y.\ \forall x\in X.\ P(x,f(x)) \)assumes \( b_2\in L_+ \) and \( b_2 + b_2 \leq b_1 \)
defines \( B_1 \equiv d^{-1}(\{c\in L^+.\ c\leq b_1\}) \) and \( B_2 \equiv d^{-1}(\{c\in L^+.\ c\leq b_2\}) \)
shows \( B_2\circ B_2 \subseteq B_1 \)assumes \( I\neq \emptyset \), \( \forall i\in I.\ A(i)\circ A(i) \subseteq B(i) \)
shows \( (\bigcap i\in I.\ A(i))\circ (\bigcap i\in I.\ A(i)) \subseteq (\bigcap i\in I.\ B(i)) \)