# Table of Contents

-   [`[2019-04-24]` charts and atlases](#chrtsndtlss) 
    -   [chart – homeomorphism from an open subset of manifold to some other space (not necessarily eucledian generally)](#chrthmmrphsmfrmnpnsbstfmnthrspcntncssrlycldngnrlly) 
    -   [atlas – collection of charts, covering the whole space](#tlscllctnfchrtscvrngthwhlspc) 
        -   [`[2019-04-24]` think of Earth as the space and atlas as a set of flat maps](#thnkfrthsthspcndtlssstffltmps) 
    -   [if codomain of atlas is eucledian, the space is a manifold](#fcdmnftlsscldnthspcsmnfld) 
        -   [local chart for manifold introduces curvilinear coordinates (coming from eucledian space)](#lclchrtfrmnfldntrdcscrvlnrcrdntscmngfrmcldnspc) 
    -   [`[2019-04-24]` https://en.wikipedia.org/wiki/Atlas\_(topology)](#snwkpdrgwktlstplgy) 
-   [`[2019-04-24]` identification of circles etc](#dntfctnfcrclstc) 
    -   [`[2019-04-24]` https://math.stackexchange.com/a/433969/15108 antipodal gluing of circle is circle again. nice gif](#smthstckxchngcmntpdlglngfcrclscrclgnncgf) 
        -   [`[2019-04-24]` antipodal identification of circle (S<sup>1</sup>) is { circle }](#ntpdldntfctnfcrclsscrcl) [[drill]]
    -   [`[2019-04-24]` identificaiton of 2D disk: right &#x2013; it's exactly the first diagram here! https://en.wikipedia.org/wiki/Real\_projective\_plane if you draw disk as a square.](#dntfctnfddskrghttsxctlythwkrlprjctvplnfydrwdskssqr) 
        -   [`[2019-04-24]` this also kinda makes sense if you draw for a bit https://math.stackexchange.com/a/1391731/15108](#thslskndmkssnsfydrwfrbtsmthstckxchngcm) 
        -   [`[2019-04-24]` antipodal identificaiton of disk (D<sup>2</sup>) is { RP<sup>2</sup> }](#ntpdldntfctnfdskdsrp) [[drill]]
        -   [`[2019-04-24]` https://mathcurve.com/surfaces.gb/planprojectif/planprojectif.shtml](#smthcrvcmsrfcsgbplnprjctfplnprjctfshtml) 
-   [`[2019-01-23]` (2) Gluing a Sphere - YouTube](#glngsphrytb) [[topology]]
-   [`[2019-01-23]` Union of two simply connected open subsets with path-connected intersection is simply connected - Topospaces](#nnftwsmplycnnctdpnsbstswtntrsctnssmplycnnctdtpspcs) [[topology]]
-   [Data type topology](#dttyptplgy) [[topology]]
    -   [`[2019-01-26]` Infinite compact sets](#nfntcmpctsts) 
    -   [`[2019-01-26]` Compactness](#cmpctnss) 
    -   [`[2019-01-26]` Equality](#qlty) 
    -   [`[2019-01-26]` Topology of Data Types](#tplgyfdttyps) 
    -   [`[2019-01-26]` References and further reading](#rfrncsndfrthrrdng) 
-   [`[2019-01-26]` A Logical Interpretation of Some Bits of Topology – XOR’s Hammer](#lgclntrprttnfsmbtsftplgyxrshmmr) [[logic]]
    -   [`[2019-04-24]` mm, not sure how this can be useful now&#x2026;](#mmntsrhwthscnbsflnw) 
-   [Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18](#twtfrmjhncrlsbzjhncrlsbztmr) 
-   [old zim notes](#ldzmnts) 
    -   [`[2016-06-18]` compactness](#cmpctnss) 
    -   [`[2016-06-20]`  connectedness](#cnnctdnss) 
    -   [`[2015-06-14]` Extracting topology from convergence](#xtrctngtplgyfrmcnvrgnc) 
-   [`[2016-06-18]` hausdorff spaces](#hsdrffspcs) [[topology]]
-   [`[2019-01-23]` (2) bothmer - YouTube](#bthmrytb) [[topology]] [[viz]] [[inspiration]]
    -   [`[2019-02-24]` some topology visualisations](#smtplgyvslstns) 
-   [`[2019-01-23]` Long line (topology) - Wikipedia](#lnglntplgywkpd) 
-   [`[2019-01-23]` N-sphere is simply connected for n greater than 1 - Topospaces](#nsphrssmplycnnctdfrngrtrthntpspcs) 
-   [`[2019-01-26]` open set = semidecidable property](#pnstsmdcdblprprty) [[drill]] [[topology]]





# `[2019-04-24]` charts and atlases





## chart &#x2013; homeomorphism from an open subset of manifold to some other space (not necessarily eucledian generally)




## atlas &#x2013; collection of charts, covering the whole space





### `[2019-04-24]` think of Earth as the space and atlas as a set of flat maps




## if codomain of atlas is eucledian, the space is a manifold





### local chart for manifold introduces curvilinear coordinates (coming from eucledian space)




## `[2019-04-24]` <https://en.wikipedia.org/wiki/Atlas_(topology)>

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.  




# `[2019-04-24]` identification of circles etc





## `[2019-04-24]` <https://math.stackexchange.com/a/433969/15108> antipodal gluing of circle is circle again. nice gif





### `[2019-04-24]` antipodal identification of circle (S<sup>1</sup>) is { circle }      [[drill]]




## `[2019-04-24]` identificaiton of 2D disk: right &#x2013; it's exactly the first diagram here! <https://en.wikipedia.org/wiki/Real_projective_plane> if you draw disk as a square.





### `[2019-04-24]` this also kinda makes sense if you draw for a bit <https://math.stackexchange.com/a/1391731/15108>




### `[2019-04-24]` antipodal identificaiton of disk (D<sup>2</sup>) is { RP<sup>2</sup> }      [[drill]]




### `[2019-04-24]` <https://mathcurve.com/surfaces.gb/planprojectif/planprojectif.shtml>

Here are classic models of the projective plane:  

-   The set of vectors of R<sup>3</sup> with the natural topology
-   A (real affine) plane completed by a projective line (line at infinity)
-   A sphere where the antipodal points are identified
-   A closed disk where the antipodal points of the circumference are identified




# `[2019-01-23]` (2) Gluing a Sphere - YouTube      [[topology]]

<https://www.youtube.com/watch?v=mmkreUEoGr8>  

    Often the fundamental group of the glued object can be calculated from the pieces (here two rectangles) and the glue (here a circle). The mathematical tool to do this is called the Seifert-van Kampen Theorem.




# `[2019-01-23]` Union of two simply connected open subsets with path-connected intersection is simply connected - Topospaces      [[topology]]

<https://topospaces.subwiki.org/wiki/Union_of_two_simply_connected_open_subsets_with_path-connected_intersection_is_simply_connected>  

    Both  and  are trivial, so we get  is an amalgamated free product of two trivial groups, hence it must be trivial.




# Data type topology      [[topology]]





## `[2019-01-26]` Infinite compact sets

<https://perl.plover.com/classes/data-topology/samples/slide022.html>  

    one-point compactification of ℕ




## `[2019-01-26]` Compactness

<https://perl.plover.com/classes/data-topology/samples/slide021.html>  

    Compact set	=	Set that can be exhaustively searched




## `[2019-01-26]` Equality

<https://perl.plover.com/classes/data-topology/samples/slide019.html>  

    Discrete space	=	Semidecidable equality




## `[2019-01-26]` Topology of Data Types

<https://perl.plover.com/classes/data-topology/>  




## `[2019-01-26]` References and further reading

<https://perl.plover.com/classes/data-topology/samples/slide027.html>  

    Other materials at http://www.cs.bham.ac.uk/~mhe/




# `[2019-01-26]` A Logical Interpretation of Some Bits of Topology – XOR’s Hammer      [[logic]]

-   State "DONE"       from              `[2019-04-24]`

<https://xorshammer.com/2011/07/09/a-logical-interpretation-of-some-bits-of-topology/>  




## `[2019-04-24]` mm, not sure how this can be useful now&#x2026;




# Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18

@zariskitopology So "compact" doesn't mean "small": it means "doesn't have any fuzzy edges".  

<https://twitter.com/johncarlosbaez/status/1106726463607209985>  




# old zim notes





## `[2016-06-18]` compactness

-   usual axioms of real numbers: forall a, b: a + b = b + a, forall x, y. exists z. x \* z > y, so on
-   add constant eps  
    -   infinite number of axioms for each n: eps < 1/n
    -   eps > 0

for each finite subset of eps axioms there clearly is a model with \bbR  
for infinite set: no model with \bbR as domain! Nonstandard real numbers, hyperreals  




## `[2016-06-20]`  connectedness

Connected: can't be represented as a union of two disjoint open sets.  
Locally connected at x: for every open V(x), there is connected open U(x) &sub; V(x). X is locally connected if locally connected at every point.  

Local connectedness and connectedness are unrelated!  

Path connected: there is a path joining every pair of points.  
Locally path connected at x: for every open V(x), there is connected open U(x) \subseteq V(x). X is locally path connected if locally path connected at every point.  

Simply connected: path-connected and fundamental group is trivial.  
Locally simply connected: admits a base of simply connected sets. Also locally path-connected and locally connected.  




## `[2015-06-14]` Extracting topology from convergence

f<sub>n</sub> -> weak(\*) f if forall x. f<sub>n</sub>(x) -> f(x)  
How to develop intuition abut the open sets?  

f<sub>n</sub> converges weakly to f if it converges pointwise  

f<sub>n</sub> converges weakly to f:  
forall O(f). exists N. forall n > N. f<sub>n</sub> &isin; O  

What is O? finite number of points do not converge?  




# `[2016-06-18]` hausdorff spaces      [[topology]]

Hausdorff if any two points can be separated by neighborhoods (diagonal is closed in product topology).  

Space X is Hausdorff iff its apartness map  

&ne; : X x X -> S  
(x, y) -> { x &ne; y }  

is continuous  

Space is discrete if every singleton is open (or if its diagonal is open)  

Space is discrete iff its equality map  

\eq : X x X -> S  
(x, y) -> { x = y }  

is continuous  




# `[2019-01-23]` (2) bothmer - YouTube      [[topology]] [[viz]] [[inspiration]]

<https://www.youtube.com/channel/UCngLGVygGfVo3pxsRzeCN_A>  




## `[2019-02-24]` some topology visualisations




# `[2019-01-23]` Long line (topology) - Wikipedia

<https://en.wikipedia.org/wiki/Long_line_(topology)>  




# `[2019-01-23]` N-sphere is simply connected for n greater than 1 - Topospaces

<https://topospaces.subwiki.org/wiki/N-sphere_is_simply_connected_for_n_greater_than_1>  




# `[2019-01-26]` open set = semidecidable property      [[drill]] [[topology]]