I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds can also be triangulated, yet the analogue of the Jordan-Schönflies Theorem in dimension 3 fails (I believe that the Alexander horned-sphere yields a counterexample). Hence, can the main idea behind the success of being able to triangulate surfaces be elucidated without fiddling with the Jordan-Schönflies Theorem?
2026-03-27 01:59:35.1774576775
Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?
196 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in MANIFOLDS
- a problem related with path lifting property
- Levi-Civita-connection of an embedded submanifold is induced by the orthogonal projection of the Levi-Civita-connection of the original manifold
- Possible condition on locally Euclidean subsets of Euclidean space to be embedded submanifold
- Using the calculus of one forms prove this identity
- "Defining a smooth structure on a topological manifold with boundary"
- On the differentiable manifold definition given by Serge Lang
- Equivalence of different "balls" in Riemannian manifold.
- Hyperboloid is a manifold
- Integration of one-form
- The graph of a smooth map is a manifold
Related Questions in SURFACES
- Surface by revolution
- A new type of curvature multivector for surfaces?
- Regular surfaces with boundary and $C^1$ domains
- Hyperplane line bundle really defined by some hyperplane
- 2D closed surface such that there's always a straight line to a point?
- parametrized surface are isometric if all corresponding curves have same length
- Klein bottle and torus in mod $p$ homology
- How can I prove that the restricted parametrization of a surface in $\mathbb{R}^{3}$ ia a diffeomorphism?
- A diffeomorphism between a cylinder and a one-sheeted hyperboloid
- Involution of the 3 and 4-holed torus and its effects on some knots and links
Related Questions in GEOMETRIC-TOPOLOGY
- Finite covers of handlebodies?
- CW complexes are compactly generated
- Constructing a fat Cantor Set with certain property
- Homologically zero circles in smooth manifolds
- Labeled graphs with unimodular adjacency matrix
- Pseudoisotopy between nonisotopic maps
- A topological question about loops and fixed points
- "Continuity" of volume function on hyperbolic tetrahedra
- Example of path connected metric space whose hyperspace with Vietoris topology is not path connected?
- What is the pushout of $D^n \longleftarrow S^{n-1} \longrightarrow D^n$?
Related Questions in TRIANGULATION
- Simplicial complex underlying space homeomorphic to torus/klein bottle
- Is the following triangulation valid?
- Need help in understanding the argument in a proof of polygon triangulation
- Triangulation of the projective plane (number of points required)
- Fake torus triangulation in Munkres
- Is this a valid triangulation of Moebius strip?
- Prove that no set of $n$ points can be triangulated in more than $2^{n \choose 2}$ ways.
- Finding the location of point P
- Cutting a square into non-similar triangles
- Delaunay Triangulation in 3D
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?