User:Alec

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!

Don't forget about: Category:Finished notes

Or User:Alec/BookTemplate

Or User:Alec/NewTodoTemplate

Or Characteristic property of the disjoint union topology/Proof

Or User:Alec/ParametricPages

Or /Modules

Or /Unnamed inequality

Or /Exam Qs

Noticeboard

As of Monday, 7/Apr/2025 at 03:22:

• All pages marked for refactoring - count: 33
  • Grade A refactoring - count: 17
  • Grade B refactoring - count: 1
  • Refactoring pages pending final review - count: 3
• All stub pages - count: 386
  • Grade A* stubs - count: 92
  • Grade A stubs - count: 118
  • Grade B stubs - count: 44
  • Stub pages pending final review - count: 2
• All pages requiring references - count: 83
  • Grade A requiring references - count: 19
  • Grade B requiring references - count: 10
  • Pages requiring references pending final review - count: 0
• Pages requiring proofs - count: 128
  • Grade A* requiring proofs - count: 8
  • Grade A requiring proofs - count: 30
  • Grade B requiring proofs - count: 14
  • Grade C requiring proofs - count: 32

Watchdogs

• Warning:There are pages (12) in Category:Refactoring (unknown grade)

• Warning:There are pages (47) in Category:Stub pages of unknown grade

• Warning:There are pages (27) in Category:Pages requiring references of unknown grade

• Warning:There are pages (25) in Category:Pages requiring proofs of unknown grade

Things not to forget

• Ensure a sigma-algebra is at the forefront of the readers' minds when dealing with measures and such @ Alec (talk) 17:45, 8 April 2016 (UTC)
• Isotonic - define a special case of monotonic where the relation is facing the same way Alec (talk) 08:44, 9 April 2016 (UTC)
  1. Possibly sorted now. See Talk:Monotonic Alec (talk) 21:19, 9 April 2016 (UTC)
• User:Alec/Things not to forget/Order theory notes Alec (talk) 20:48, 9 April 2016 (UTC)
• Greater than or equal to - I made a (non-destructive) mistake in the proof, meaning my result is too concrete (as opposed to abstract). Don't forget to fix Alec (talk) 21:19, 9 April 2016 (UTC)
• Work on Ordered pair (lacks references) and
• Work on Cardinality (incomplete, not marked as stub....)
• Create some notion of urgency with regards to requiring references or proofs. Also maybe tag "low-hanging fruit" (easy proofs) so a new-comer has something they can do.
• Add that new Order Theory book.
• Bounded (ordering)
• The fundamental group of the product of a finite number of topological spaces is the product of the fundamental groups of each
  • Investigate the arbitrary and countably infinite cases. I suspect the product of groups drops to the direct sum of the groups.
• An $\epsilon$-$\delta$ form of continuity is missing.
• Partial orderings are to total orderings what partial function is to total function. That's where the partial comes from!
• Semi-ring pre-measure vs Pre-measure on a semi-ring Alec (talk) 00:31, 20 August 2016 (UTC)
• "Universal Mapping Property" = Group homomorphism theorem according to McCoy, Algebra: Groups, rings, and other topics - page 69 - Alec (talk) 20:29, 4 October 2016 (UTC)
• User:Alec/Things not to forget/Q9 wreckage

Pages being worked on

• Outer-measure @ Alec (talk) 17:45, 8 April 2016 (UTC)

Notation

Symbol	Command
$\trianglelefteq$	\trianglelefteq
$\triangleq$	\triangleq
$\ntrianglelefteq$	\ntrianglelefteq

Current projects

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

• Notes:Halmos measure theory skeleton - notes for plan on definitions
• Site projects:Patrolling measure theory - "patrol" being the "decent standard" project.
• Site projects:Measure theory plan - implementation of plan.
• Site projects:Split set theory into elementary set theory - some things classified as "set theory" (like set subtraction) would be best classified as "elementary set theory". This project endeavorers to do that.
• Site projects:Patrolling topology - "patrol" being the "decent standard" project.

Notes

Diagram for Characteristic property of the product topology:

$\begin{xy} \xymatrix{ & & \prod_{\alpha\in I}X_\alpha \ar[dd] \\ & & \\ Y \ar[uurr]^f \ar[rr]+<-0.9ex,0.15ex>|(.875){\hole} & & X_b \save (15,13)+"3,3"*+{\ldots}="udots"; (8.125,6.5)+"3,3"*+{X_a}="x1"; (-8.125,-6.5)+"3,3"*+{X_c}="x3"; (-15,-13)+"3,3"*+{\ldots}="ldots"; \ar@{->} "x1"; "1,3"; \ar@{->}_(0.55){\pi_c,\ \pi_b,\ \pi_a} "x3"; "1,3"; \ar@{->}|(.873){\hole} "x1"+<-0.9ex,0.15ex>; "3,1"; \ar@{->}_{f_c,\ f_b,\ f_a} "x3"+<-0.9ex,0.3ex>; "3,1"; \restore } \end{xy}$

Diagram

Characteristic property of the subspace topology:

• $$\forall S\in\mathcal{P}(X)\left[\forall Y\forall\mathcal{K}\in\mathcal{P}(\mathcal{P}(Y))\left(\overbrace{\mathcal{T}(Y,\mathcal{K})\implies}^{\text{if }(Y,\mathcal{K})\text{ is a topology then} }\Big[\underbrace{\forall f\in\mathcal{P}(Y\times X)}_{\text{relations from }X\text{ to }Y}\big(\overbrace{\mathcal{F}(f)\implies}^{\begin{array}{c}\text{if }f\text{ is a}\\\text{function then} \end{array} }[\mathcal{C}(f)\iff\mathcal{C}(\iota_S\circ f)]\big)\Big]\right)\right]$$

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.
             */