📚 node [[topology]]
📓 garden/flancian/topology.md by @flancian ️👁 ️📝

topology
📓 garden/karlicoss/math/topology.md by @karlicoss

Table of Contents

[2019-04-24] charts and atlases

chart – homeomorphism from an open subset of manifold to some other space (not necessarily eucledian generally)

atlas – 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 (S1) is { circle } [[drill]]

[2019-04-24] identificaiton of 2D disk: right – 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 (D2) is { RP2 } [[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 R3 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…

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) ⊂ 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

fn -> weak(*) f if forall x. fn(x) -> f(x)
How to develop intuition abut the open sets?

fn converges weakly to f if it converges pointwise

fn converges weakly to f:
forall O(f). exists N. forall n > N. fn ∈ 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

≠ : X x X -> S
(x, y) -> { x ≠ 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]]
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