[2019-04-24]
charts and atlases [2019-04-24]
https://en.wikipedia.org/wiki/Atlas_(topology) [2019-04-24]
identification of circles etc
[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]
A Logical Interpretation of Some Bits of Topology β XORβs Hammer [[logic]]
[2016-06-18]
hausdorff spaces [[topology]][2019-01-23]
(2) bothmer - YouTube [[topology]] [[viz]] [[inspiration]]
[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[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 β 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.shtmlHere are classic models of the projective plane:
[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]][2016-06-18]
compactnessfor 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]
connectednessConnected: 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) β 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 convergencefn -> 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 β 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
β : X x X -> S
(x, y) -> { x β 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
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