General topology

Table of Contents

[2019-04-24] charts and atlases mathtopology

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

atlas – collection of charts, covering the whole space mathtopology

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

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

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

[2019-04-24] mathtopology

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 mathtopology

[2019-04-24] antipodal gluing of circle is circle again. nice gif mathtopology

[2019-04-24] antipodal identification of circle (S1) is { circle } mathtopologydrill

[2019-04-24] identificaiton of 2D disk: right – it's exactly the first diagram here! if you draw disk as a square. mathtopology

[2019-04-24] this also kinda makes sense if you draw for a bit mathtopology

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

[2019-04-24] mathtopology

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

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

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.

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

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

[D] Data type topology mathtopology

[C] [2019-01-26] Infinite compact sets mathtopology

[C] [2019-01-26] Compactness mathtopology

Compact set	=	Set that can be exhaustively searched

[C] [2019-01-26] Equality mathtopology

[C] [2019-01-26] Topology of Data Types mathtopology

[C] [2019-01-26] References and further reading mathtopology

Other materials at

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

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

TODO [D] Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18 mathtopology

CREATED: [2019-03-16]

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

TODO [C] old zim notes mathtopology

[2016-06-18] compactness mathtopology

  • 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 mathtopology

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 mathtopology

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 mathtopology

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

[C] [2019-01-23] (2) bothmer - YouTube mathtopologyvizinspiration

[2019-02-24] some topology visualisations mathtopologyvizinspiration

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

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

[2019-01-26] open set = semidecidable property mathtopologydrill

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