Difference between revisions of "User:Alec"

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Characteristic property of the subspace topology:
 
Characteristic property of the subspace topology:
 
* {{MM|1=\forall S\in\mathcal{P}(X)\left[\forall Y\forall\mathcal{K}\in\mathcal{P}(\mathcal{P}(Y))\left(\overbrace{\mathcal{T}(Y,\mathcal{K})\implies}^{\text{if }(Y,\mathcal{K})\text{ is a topology then} }\Big[\underbrace{\forall f\in\mathcal{P}(Y\times X)}_{\text{relations from }X\text{ to }Y}\big(\overbrace{\mathcal{F}(f)\implies}^{if f is afunction then
}[\mathcal{C}(f)\iff\mathcal{C}(\iota_S\circ f)]\big)\Big]\right)\right]}}
 
* {{MM|1=\forall S\in\mathcal{P}(X)\left[\forall Y\forall\mathcal{K}\in\mathcal{P}(\mathcal{P}(Y))\left(\overbrace{\mathcal{T}(Y,\mathcal{K})\implies}^{\text{if }(Y,\mathcal{K})\text{ is a topology then} }\Big[\underbrace{\forall f\in\mathcal{P}(Y\times X)}_{\text{relations from }X\text{ to }Y}\big(\overbrace{\mathcal{F}(f)\implies}^{if f is afunction then
}[\mathcal{C}(f)\iff\mathcal{C}(\iota_S\circ f)]\big)\Big]\right)\right]}}
 +
==Things I want to use==
 +
Linked list documentation diagram:
 +
<syntaxhighlight lang="c">
 +
            /*
 +
            * B is the node to be removed.
 +
            * We do this by binding A forward to C
 +
            *
 +
            * BEFORE:
 +
            * ---> +---+---------> +---+---------> +---+-----
 +
            *....  | A |          | B |          | C |    ....
 +
            * -----+---+ <---------+---+ <---------+---+ <---
 +
            *
 +
            * AFTER:      ,------------------.
 +
            * ---> +---+--'        +---+-------'-> +---+-----
 +
            *....  | A |          | B |          | C |    ....
 +
            * -----+---+ <-.-------+---+        ,--+---+ <---
 +
            *              `------------------'
 +
            *
 +
            *  Note that B.previous.bindForwards(B.next) is the same
 +
            *  as B.next.bindBackwards(B.previous)
 +
            *  A nice side-effect of this is that provided one has a
 +
            *  ref counting scheme in place (which you need basically
 +
            *  need for safe multi-threaded schemes anyway) any iterators
 +
            *  "pointing" to B remain valid.
 +
            *  Make sure that this "rail-road" situation where going
 +
            *  forward and back MIGHT not be inverses is well documented
 +
            *  and expected.
 +
            */
 +
</syntaxhighlight>

Revision as of 08:42, 8 June 2016

Hi, I'm Alec Teal; the admin of this site and creator of most of the content. Write a note on my talk page for requests!

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Currently ensuring that all existing measure theory articles are "up to a decent standard" and working on Measure Theory, following Halmos' example (rings).

Notes

Diagram for Characteristic property of the product topology:

Diagram

Characteristic property of the subspace topology:

  • ∀S∈P(X)[∀Y∀K∈P(P(Y))(if (Y,K) is a topology then⏞T(Y,K)⟹[∀f∈P(Y×X)⏟relations from X to Y(if f is afunction then⏞F(f)⟹[C(f)⟺C(ιS∘f)])])]

Things I want to use

Linked list documentation diagram: 

            /*
             * B is the node to be removed.
             * We do this by binding A forward to C
             * 
             * BEFORE: 
             * ---> +---+---------> +---+---------> +---+-----
             *....  | A |           | B |           | C |    ....
             * -----+---+ <---------+---+ <---------+---+ <---
             * 
             * AFTER:       ,------------------.
             * ---> +---+--'        +---+-------'-> +---+-----
             *....  | A |           | B |           | C |    ....
             * -----+---+ <-.-------+---+        ,--+---+ <---
             *               `------------------'
             * 
             *  Note that B.previous.bindForwards(B.next) is the same
             *  as B.next.bindBackwards(B.previous)
             *  A nice side-effect of this is that provided one has a 
             *  ref counting scheme in place (which you need basically
             *  need for safe multi-threaded schemes anyway) any iterators
             *  "pointing" to B remain valid.
             *  Make sure that this "rail-road" situation where going
             *  forward and back MIGHT not be inverses is well documented
             *  and expected.
             */
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