Showing posts with label Definitions and Propositions. Show all posts
Showing posts with label Definitions and Propositions. Show all posts

2024-04-14

538: Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space

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definition of convex set spanned by possibly-non-affine-independent set of base points on real vectors space

Topics


About: vectors space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of convex set spanned by possibly-non-affine-independent set of base points on real vectors space.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
V: { the real vectors spaces }
{p0,...,pn}: V, { the possibly-non-affine-independent sets of base points on V}
S: ={j=0ntjpjV|tjR,j=0ntj=10tj}
//

Conditions:
//


2: Natural Language Description


For any real vectors space, V, and any possibly-non-affine-independent set of base points, p0,...,pnV, the set, S:={j=0ntjpjV|tjR,j=0ntj=10tj}, which is the set of all the convex combinations of the set of the base points


3: Note


S is not necessarily any affine simplex spanned by an affine-independent subset of the base points, by the proposition that the convex set spanned by a non-affine-independent set of base points on a real vectors space is not necessarily any affine simplex spanned by an affine-independent subset of the base points.

But S is a convex set anyway, as is proved in the proposition that the convex set spanned by any possibly-non-affine-independent set of base points on any real vectors space is convex.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

537: Affine Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of affine set spanned by possibly-non-affine-independent set of base points on real vectors space

Topics


About: vectors space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of affine set spanned by possibly-non-affine-independent set of base points on real vectors space.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
V:  the real vectors spaces 
{p0,...,pn}: V, { the possibly-non-affine-independent sets of base points on V}
S: {j=0ntjpjV|tjR,j=0ntj=1}
//

Conditions:
//


2: Natural Language Description


For any real vectors space, V, and any possibly-non-affine-independent set of base points, p0,...,pnV, the set, S:={j=0ntjpjV|tjR,j=0ntj=1}, which is the set of all the affine combinations of the set of the base points


3: Note


S is the affine set spanned by an affine-independent subset of the base points, by the proposition that the affine set spanned by any non-affine-independent set of base points on any real vectors space is the affine set spanned by an affine-independent subset of the base points.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

536: Convex Combination of Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of convex combination of possibly-non-affine-independent set of base points on real vectors space

Topics


About: vectors space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of convex combination of possibly-non-affine-independent set of base points on real vectors space.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
V:  the real vectors spaces 
{p0,...,pn}: V, { the possibly-non-affine-independent sets of base points on V}
p: =j=0ntjpjV, tjR
//

Conditions:
j=0ntj=10tj
//


2: Natural Language Description


For any real vectors space, V, and any possibly-non-affine-independent set of base points, p0,...,pnV, any point, p=j=0ntjpjV, such that tjR,j=0ntj=10tj


References


<The previous article in this series | The table of contents of this series | The next article in this series>

535: Affine Combination of Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of affine combination of possibly-non-affine-independent set of base points on real vectors space

Topics


About: vectors space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of affine combination of possibly-non-affine-independent set of base points on real vectors space.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
V:  the real vectors spaces 
{p0,...,pn}: V, { the possibly-non-affine-independent sets of base points on V}
p: =j=0ntjpjV, tjR
//

Conditions:
j=0ntj=1
//


2: Natural Language Description


For any real vectors space, V, and any possibly-non-affine-independent set of base points, p0,...,pnV, any point, p=j=0ntjpjV, such that tjR,j=0ntj=1


References


<The previous article in this series | The table of contents of this series | The next article in this series>

534: Affine-Independent Set of Points on Real Vectors Space

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of affine-independent set of points on real vectors space

Topics


About: vectors space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of affine-independent set of points on real vectors space.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
V:  the real vectors spaces 
{p0,...,pn}: V
//

Conditions:
j{0,...,n}({p0pj,...,pjpj^,...,pnpj} is linearly independent ), where the hat mark denotes that the component is missing.
//


2: Natural Language Description


For any real vectors space, V, any set of points, {p0,...,pn}V, such that for each pj, {p0pj,...,pjpj^,...,pnpj} is linearly independent, where the hat mark denotes that the component is missing


3: Note


In fact, if the set is linearly independent for a pj, inevitably, the set is linearly independent for each pj, by the proposition that for any finite set of points on any real vectors space, if for one of the points, the set of the subtractions of the point from the other points is linearly independent, it is so for each point.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

533: For Finite Set of Points on Real Vectors Space, if for Point, Set of Subtractions of Point from Other Points Is Linearly Independent, It Is So for Each Point

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that for finite set of points on real vectors space, if for point, set of subtractions of point from other points are linearly independent, it is so for each point

Topics


About: vectors space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any finite set of points on any real vectors space, if for one of the points, the set of the subtractions of the point from the other points is linearly independent, it is so for each point.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
V: { the real vectors spaces }
{p0,...,pn}: V
//

Statements:
pk({p0pk,...,pkpk^,pnpk} is linearly independent ), where the hat mark denotes that the component is missing

pl({p0pl,...,plpl^,pnpl} is linearly independent ).
//


2: Natural Language Description


For any real vectors space, V, and any finite set of points, {p0,...,pn}V, if for a pk, {p0pk,...,pjpk^,pnpk} is linearly independent, where the hat mark denotes that the component is missing, for each pl, {p0pl,...,plpl^,pnpl} is linearly independent.


3: Proof


Let us suppose that {p0pk,...,pkpk^,pnpk} is linearly independent.

Let l be any l{0,...,n} and lk and Jl:={0,...,n}{l}.

Let jJl(tj(pjpl))=0. =jJl(tj(pjpk+pkpl))=jJl(tj(pjpk))jJl(tj(plpk))=jJl(tj(pjpk))(jJltj)(plpk). Then, tj=0 for each jk and jJltj=0, but jJltj=tk=0. So, each tj is 0, which means that {p0pl,...,plpl^,pnpl} is linearly independent.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

532: Linearly Independent Subset of Module

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of linearly independent subset of module

Topics


About: module

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of linearly independent subset of module.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
R: { the rings }
M: { the R modules }
S: M, { the possibly uncountable sets }
//

Conditions:
SS,S{ the finite sets }
(
pjSrjpj=0,rjR

j(rj=0)
)
//


2: Natural Language Description


For any module, M, over any ring, R, any (possibly uncountable) subset, SM, such that for each finite subset, SS, pjSrjpj=0 implies that rj=0 for each j, where rj is any element of R


3: Note


As any vectors space is a module, 'linearly independent subset of vectors space' is nothing but 'linearly independent subset of module'.

We need to think of the linear combination of each finite subset instead of the linear combination of possibly infinite S, because the convergence of the linear combination of any infinite elements is not defined without any extra structure like topology or metric.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

531: %Ring Name% Module

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of %ring name% module

Topics


About: module

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of %ring name% module.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
R: { the rings }
M: { the sets } with any +:M×MM (addition) operation and any .:R×MM (scalar multiplication) operation
//

Conditions:
1) m1,m2M(m1+m2M) (closed-ness under addition)

2) m1,m2M(m1+m2=m2+m1) (commutativity of addition)

3) m1,m2,m3M((m1+m2)+m3=m1+(m2+m3)) (associativity of additions)

4) 0M(mM(m+0=m)) (existence of 0 element)

5) mM(mM(m+m=0)) (existence of inverse vector)

6) mM,rR(r.mM) (closed-ness under scalar multiplication)

7) mM,r1,r2R((r1+r2).m=r1.m+r2.m) (scalar multiplication distributability for scalars addition)

8) m1,m2M,rR(r.(m1+m2)=r.m1+r.m2) (scalar multiplication distributability for elements addition)

9) mM,r1,r2R((r1r2).m=r1.(r2.m)) (associativity of scalar multiplications)

10) mM(1.m=m) (identity of 1 multiplication)
//


2: Natural Language Description


Any set, M, with any +:M×MM (addition) operation and any .:R×MM (scaler multiplication) operation with respect to any ring, R, that satisfies these conditions: 1) for any elements, m1,m2M, m1+m2M (closed-ness under addition); 2) for any elements, m1,m2M, m1+m2=m2+m1 (commutativity of addition); 3) for any elements, m1,m2,m3M, (m1+m2)+m3=m1+(m2+m3) (associativity of additions); 4) there is a 0 element, 0M, such that for each mM, m+0=m (existence of 0 element); 5) for any element, mM, there is an inverse element, mM, such that m+m=0 (existence of inverse vector); 6) for any element, mM, and any scalar, rR, r.mM (closed-ness under scalar multiplication); 7) for any element, mM, and any scalars, r1,r2R, (r1+r2).m=r1.m+r2.m (scalar multiplication distributability for scalars addition); 8) for any elements, m1,m2M, and any scalar, rR, r.(m1+m2)=r.m1+r.m2 (scalar multiplication distributability for elements addition); 9) for any element, mM, and any scalars, r1,r2R, (r1r2).m=r1.(r2.m) (associativity of scalar multiplications); 10) for any element, mM, 1.m=m (identity of 1 multiplication)


3: Note


. is often omitted in notations like rm instead of r.m.

The requirements 1) ~ 10) are in fact parallel to those for vectors space; the difference between vectors space and module is only that the scalar structure is a field or a ring, which makes a significant difference, because for example, for a module, r1m1+r2m2+r3m3=0 where r10 does not imply that m1 is a linear combination of m2 and m3, because r11 is not guaranteed to exist.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

2024-04-07

524: Product Map

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of product map

Topics


About: set

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of product map.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description 1


Here is the rules of Structured Description.

Entities:
A: { the possibly uncountable index sets }
{Sα}: αA, Sα{ the sets }
{Sα}: αA, Sα{ the sets }
{fα}: αA, :SαSα
×αAfα: :×αASα×αASα,(αf(α))(αfα(f(α)))
//

Conditions:
//


2: Natural Language Description 1


For any possibly uncountable index set, A, any sets, {Sα|αA}, any sets, {Sα|αA}, and any maps, {fα:SαSα}, ×αAfα:×αASα×αASα, (αf(α))(f:αfα(f(α)))


3: Structured Description 2


Here is the rules of Structured Description.

Entities:
J: ={1,...,n}
{Sj}: jJ, Sj{ the sets }
{Sj}: jJ, Sj{ the sets }
{fj}: jJ, :SjSj
f1×f2×...×fn: :S1×S2×...×SnS1×S2×...×Sn,(p1,p2,...,pn)(f1(p1),f2(p2),...,fn(pn))


Conditions:
//


4: Natural Language Description 2


For any finite number of sets, S1,S2,...,Sn, any same number of sets, S1,S2,...,Sn, and any same number of maps, f1:S1S1,f2:S2S2,...,fn:SnSn, f1×f2×...×fn:S1×S2×...×SnS1×S2×...×Sn, (p1,p2,...,pn)(f1(p1),f2(p2),...,fn(pn))


References


<The previous article in this series | The table of contents of this series | The next article in this series>

530: Sufficient Conditions for Existence of Unique Global Solution on Interval for Euclidean-Normed Euclidean Vectors Space ODE

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of sufficient conditions for existence of unique global solution on interval for Euclidean-normed Euclidean vectors space ODE

Topics


About: normed vectors space
About: differential equation

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of some sufficient conditions with which there is the unique solution on the whole interval with any initial condition for any Euclidean-normed Euclidean vectors space ordinary differential equation.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description 1


Here is the rules of Structured Description.

Entities:
Rd: = the Euclidean-normed Euclidean vectors space 
R: = the Euclidean-normed Euclidean vectors space 
J: R, =[t1,te]
f: :Rd×JRd, { the C0 maps }
dxdt=f(x,t): = the ordinary differential equation with any initial condition, x(t1)=xt1
//

Statements:
t0J
(
[t0ϵt0,1,t0+ϵt0,2] that satisfies the conditions for the local solution existence of the equation for the initial condition with any initial value, xt0, for t0, where ϵt0,j is not zero except ϵt1,1 and ϵte,2 and does not depend on xt0 although can depend on t0, which means that there is a map, xt0x(t) for each t[t0ϵt0,1,t0+ϵt0,2], and the map is bijective
)

The ordinary differential equation has the unique solution on the entire J.
//


2: Natural Language Description 1


For any Euclidean-normed Euclidean vectors space, Rd and R, any closed interval, JR=[t1,te], and any C0 map, f:Rd×JRd, the ordinary differential equation, dxdt=f(x,t) with any initial condition x(t1)=xt1, has the unique solution on the entire J if at each point, t0J, there is a closed interval, [t0ϵt0,1,t0+ϵt0,2], that satisfies the conditions for the local solution existence of the equation for the initial condition with any initial value, xt0, for t0, where ϵt0,j is not zero except ϵt1,1 and ϵte,2, and does not depend on xt0 although can depend on t0, which means that there is a map, xt0x(t) for each t[t0ϵt0,1,t0+ϵt0,2], and the map is bijective.


3: Proof 1


The open intervals, (t0ϵt0,1,t0+ϵt0,2) for t0t1,te, [t1,t1+ϵt1,2), and (teϵte,1,te], cover compact J, so, there is a finite subcover, [t1,t1+ϵt1,2),(t2ϵt2,1,t2+ϵt2,2),...,(teϵte,1,te].

There is the local unique solution for [t1,t1+ϵt1,2) with the initial condition, x(t1)=xt1, x:[t1,t1+ϵt1,2)Rd. Let us define the restriction of x, x|[t1,t1+ϵt1,2)=x.

There is an interval that intersects [t1,t1+ϵt1,2), (tj2ϵtj2,1,tj2+ϵtj2,2), with an intersection point denoted as t1,j2, and there is the local unique solution for it with the initial condition, x(tj2)=xtj2, where xtj2 can be chosen such that x(t1,j2)=x(t1,j2), which is because the bijective map mentioned in Description is supposed to exist. In fact, x coincides with x on the whole intersection area, because there is an interval around t1,j2 and there is the unique solution on it with the initial condition, which has to agree with x and x there, and if x and x did not agree on the whole intersection area, (tj2ϵtj2,1,tj2+ϵtj2,2) would have another solution by switching to x after passing t1,j2 or [t1,t1+ϵt1,2) would have another solution by switching to x after passing t1,j2. Let us extend x to [t1,tj2+ϵtj2,2) with x.

Thus, x can be extended to the whole J. The extension is unique, because at each point, tjk, xtjk is determined uniquely.


4: Note 1


It is crucial that such a bijective map exists for this proposition. Just that there is a local solution around each point does not guaranteed the existence of any global solution, as is described in another article.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

529: Restriction of Continuous Embedding on Domain and Codomain Is Continuous Embedding

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that restriction of continuous embedding on domain and codomain is continuous embedding

Topics


About: topological space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that any restriction of any continuous embedding on the domain and the codomain is a continuous embedding.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
T1: { the topological spaces }
T2: { the topological spaces }
f: :T1T2, { the continuous embeddings }
S1: T1
S2: T2, f(S1)S2
f: :S1S2,pf(p)
//

Statements:
f{ the continuous embeddings }
//


2: Natural Language Description


For any topological spaces, T1,T2, any continuous embedding, f:T1T2, any subset, S1T1, and any subset, S2T2 such that f(S1)S2, f:S1S2,pf(p) is a continuous embedding.


3: Proof


f is injective, because f is so.

f is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.

Let us denote the codomain restriction of f as f:S1f(S1).

f is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.

As f is bijective, there is the inverse, f1:f(S1)S1. Is f1 continuous?

For any open subset, US1, is f11(U)=f(U) open on f(S1)?

U=US1 for an open subset, UT1. f(U)=f(US1)=f(U)f(S1), by the proposition that for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets, =f(U)f(S1)=Uf(T1)f(S1) where UT2 is an open subset, because f:T1f(T1) is a homeomorphism (so, f(U) is open on f(T1)), =Uf(S1), which is open on f(S1)=f(S1).

So, yes, f1 is continuous, and f is a homeomorphism.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

528: For Infinite Product Topological Space and Closed Subset, Point on Product Space Whose Each Finite-Components-Projection Belongs to Corresponding Projection of Subset Belongs to Subset

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that for infinite product topological space and closed subset, point on product space whose each finite-components-projection belongs to corresponding projection of subset belongs to subset

Topics


About: topological space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any infinite product topological space and any closed subset, any point on the product space whose each finite-components-projection belongs to the corresponding projection of the subset belongs to the subset.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
A: { the possibly uncountable infinite index sets }
{Tα|αA}: Tα{ the topological spaces }
T: =×αATα with the product topology
C: T, { the closed subsets }
p: T
//

Statements:
JA,J{ the finite index sets }(πJ(p)πJ(C)), where πJ:T×jJTj is the projection

pC
//


2: Natural Language Description


For any possibly uncountable infinite index set, A, any topological spaces, {Tα|αA}, the product topological space, T:=×αATα, any closed CT, and any pT, πJ(p)πJ(C) for each JA, where J is a finite index set and πJ:T×jJTj is the projection, implies that pC.


3: Note 1


A typical case is T=T1×T2×..., a infinitely countable product, and πJ(p)πJ(C) where J={1,...,k} for each k implies pC: the requirement for such Js implies the requirement for each JJ, so, implies the requirement for each finite JA.


4: Proof


Let us suppose that πJ(p)πJ(C) for each J.

TCT is open.

Let us suppose that pC.

pTC. There would be an open neighborhood, UpTC, of p. Up=βB×αAUβ,α, where B would be a possibly uncountable index set and Uβ,αTα would be open while only finite number of Uβ,α s would not be Tα s for each β (see Note of the definition of product topology).

There would be a β such that p×αAUβ,αTC. As only finite of Uβ,α s would not be Tα s, let J be of the finite components. πJ(p)πJ(×αAUβ,α)=×jJUβ,j. Then, there would be a pC such that πJ(p)=πJ(p)×jJUβ,j. But p×αAUβ,α, because Uβ,α=Tα for each αJ and so, pαUβ,α when αJ and when αAJ.

So, p×αAUβ,αβB×αAUβ,α=UpTC, which would mean pC, a contradiction.

So, pC.


5: Note 2


When CT is not closed, pC is not necessarily implied as is proved in another article.


References


<The previous article in this series | The table of contents of this series | The next article in this series>

527: For Infinite Product Topological Space and Subset, Point on Product Space Whose Each Finite-Components-Projection Belongs to Corresponding Projection of Subset Does Not Necessarily Belong to Subset

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that for infinite product topological space and subset, point on product space whose each finite-components-projection belongs to corresponding projection of subset does not necessarily belong to subset

Topics


About: topological space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for an infinite product topological space and a subset, a point on the product space whose each finite-components-projection belongs to the corresponding projection of the subset does not necessarily belong to the subset.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
A: { the possibly uncountable infinite index sets }
{Tα|αA}: Tα{ the topological spaces }
T: =×αATα with the product topology
S: T
p: T
//

Statements:
JA,J{ the finite index sets }(πJ(p)πJ(S)) does not necessarily imply pS, where πJ:T×jJTj is the projection.
//


2: Natural Language Description


For any possibly uncountable infinite index set, A, any topological spaces, {Tα|αA}, the product topological space, T:=×αATα, any ST, and any pT, πJ(p)πJ(S) for each JA, where J is a finite index set and πJ:T×jJTj is the projection, does not necessarily imply that pS.


3: Note 1


A typical case is T=T1×T2×..., a infinitely countable product, and πJ(p)πJ(S) where J={1,...,k} for each k does not imply pS.


4: Proof


A counterexample suffices.

Let each Tα have at least 2 points, pα,1,pα,2Tα, let S=T×αA{pα,1}, and let each α component of p be pα,1.

pT. πJ(p)πJ(S) for each J, because S has at least 1 point, p, whose J components are pj,1 s (jJ) but whose α component is pα,2 where αJ; pS, because pα,2{pα,1}.

But pS, because p×αA{pα,1}.

Letting A be countable does not help. Let A={1,2,...}, let each Tj have at least 2 points, pj,1,pj,2Tj, let S=T({p1,1}×{p2,1}×...), and let p be (p1,1,p2,1,...).

pT. πJ(p)πJ(S)) for each J, because, for example, when J={2,4}, (p2,2,p2,1,p3,1,p4,1,...),(p1,1,p2,1,p3,2,p4,1,...),...S, while πJ(p)=πJ((p2,2,p2,1,p3,1,p4,1,...)))=(p2,1,p4,1).

But pS, because p{p1,1}×{p2,1}×....


5: Note 2


When ST is closed, pS is implied as is proved in another article.


References


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