📚 Node [[topology]] exact match ☆

Nodes contain individual contributions whose filenames match your search. x

📓 topology.md by @flancian ☆ ️🔗 ✍️

topology

the study of [[shapes]] and [[spaces]]
- [[caveat emptor]] my definition

📓 math/topology.md by @karlicoss ☆

[2019-04-24] charts and atlases
[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.
[2019-01-23] (2) Gluing a Sphere - YouTube [[topology]]
[2019-01-23] Union of two simply connected open subsets with path-connected intersection is simply connected - Topospaces [[topology]]
[[Data type topology]] [[topology]]
[2019-01-26] A Logical Interpretation of Some Bits of Topology – XOR’s Hammer [[logic]]
- [2019-04-24] mm, not sure how this can be useful now…
[[Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18]]
[[old zim notes]]
[2016-06-18] hausdorff spaces [[topology]]
[2019-01-23] (2) bothmer - YouTube [[topology]] [[viz]] [[inspiration]]
- [2019-02-24] some topology visualisations
[2019-01-23] Long line (topology) - Wikipedia
[2019-01-23] N-sphere is simply connected for n greater than 1 - Topospaces
[2019-01-26] open set = semidecidable property [[drill]] [[topology]]

`[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)

`[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

`[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 >

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

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

f_n converges weakly to f if it converges pointwise

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

📓 Topology.md by @agora@botsin.space ☆

[[jonny@neuromatch.social]] https://neuromatch.social/@jonny/109707657307681691

Loading pushes...

Rendering context...

📚 Node [[network topology]] perhaps related

📚 Node [[neuraltopology]] perhaps related

📚 Node [[synthetic topology]] perhaps related

📚 Node [[the topology of information networks]] perhaps related

topology

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)

`[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

`[2019-04-24]` antipodal identificaiton of disk (D²) is { RP² } [[drill]]

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

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

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

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

`[2019-01-26]` Compactness

`[2019-01-26]` Equality

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

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

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

`[2019-04-24]` mm, not sure how this can be useful now…

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

old zim notes

`[2016-06-18]` compactness

`[2016-06-20]` connectedness

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

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

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

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

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

topology

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) 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 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

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

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

[2019-01-26] Infinite compact sets

[2019-01-26] Compactness

[2019-01-26] Equality

[2019-01-26] Topology of Data Types

[2019-01-26] References and further reading

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

[2019-04-24] mm, not sure how this can be useful now…

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

old zim notes

[2016-06-18] compactness

[2016-06-20] connectedness

[2015-06-14] Extracting topology from convergence

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

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

[2019-02-24] some topology visualisations

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

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

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

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

`[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

`[2019-04-24]` antipodal identificaiton of disk (D²) is { RP² } [[drill]]

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

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

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

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

`[2019-01-26]` Compactness

`[2019-01-26]` Equality

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

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

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

`[2019-04-24]` mm, not sure how this can be useful now…

`[2016-06-18]` compactness

`[2016-06-20]` connectedness

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

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

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

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

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