Intereting Posts

How many nodes are there in a 5-regular planar graph with diameter 2?
Power series without analytic continuation
Separability of a product metric space
How to derive the gregory series for inverse tangent function?
Proving that $ f: \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.
Minimum number of edge-disjoint paths needed to cover a graph
Show that $R/(I \cap J) \cong (R/I) \times (R/J) $
explaining the derivative of $x^x$
Show that $\lim\limits_{x\to0}\frac{e^x-1}{x}=1$
Does this integral have a closed form: $\int_0^1 \frac{x^{\beta-1}}{1-x}\log\frac{1-y x^\delta}{1-y}\mathrm dx$?
Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions
How do we know that $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ isn't closed?
Book about the foundation of math?
A good reference on optimal control theory
Help with proof of Jensens inequality

I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?

Moreover, is there a theorem which classifies the 2-manifolds with boundary?

- For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?
- Properly discontinuous action: equivalent definitions
- Homotopically trivial $2$-sphere on $3$-manifold
- First Pontryagin class on real Grassmannian manifold?
- A manifold such that its boundary is a deformation retract of the manifold itself.
- $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, what is $\omega$?

- $(0,1)$ is an open subset of $\mathbb{R}$ but not of $\mathbb{R}^2$, when we think of $\mathbb{R}$ as the x-axis in $\mathbb{R}^2$. Prove this.
- Inverse image of a compact set is compact
- Urysohn's lemma with Lipschitz functions
- What is $(S^1\times S^1)/C_{n}$ topologically?
- $\forall A\subset \mathbb{N}$ the sum of the reciprocals of $A$ diverges iff $A$ is $(\tau, \mathbb{N})$-dense
- Showing a diffeomorphism extends to the neighborhood of a submanifold
- When is there a submersion from a sphere into a sphere?
- When is a notion of convergence induced by a topology?
- Can $k$ be dense in $k$? where $p_xq_y-p_yq_x \in k^*$.
- Error in book's definition of open sets in terms of neighborhoods?

Yes, there’s a classification theorem for non-compact 2-manifolds.

This paper gives the classification for triangulable 2-manifolds:

http://www.jstor.org/stable/1993768

That an arbitrary (2nd countable, Hausdorff) topological 2-manifold admits a triangulation is fairly classical. Ahlfors book “Riemann Surfaces” has a proof. There are others available, see for example this list:

https://mathoverflow.net/questions/17578/triangulating-surfaces

If all you’re interested in is compact manifolds with boundary, you get that classification immediately from the closed manifold case. Because if you have a compact manifold with boundary, its boundary is a disjoint union of circles. So cap those circles off with discs to produce a closed manifold. So compact manifolds with boundary are classified by the closed manifold you get by “capping off” and the number of boundary circles you started with.

Non-compact manifolds have a more delicate classification — think for example about the complement of a Cantor set in a compact surface.

Amazingly, the complete classification of noncompact 2-manifolds with boundary was not completed until 2007. Here’s the reference:

A. O. Prishlyak and K. I. Mischenko, Classification of noncompact surfaces with

boundary, Methods Funct. Anal. Topology 13 (2007), no. 1, 62–66.

- Computing kernel
- Why did no student get the correct answer?
- Find all integral solutions to $a+b+c=abc$.
- Is $\sqrt{p+q\sqrt{3}}+\sqrt{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?
- elementary prove thru induction – dumb stumbling
- Darboux Theorem
- $n\times n$ chessboard game with coins
- If a and b are non-negative real numbers then demonstrate inequality
- inductive proof for $\binom{2n}{n}$
- Proof of equicontinuous and pointwise bounded implies compact
- How calculators do trigonometry
- How to show that $\sqrt{2}+\sqrt{3}$ is algebraic?
- Show that $y = \frac{2x}{x^2 +1}$ lies between $-1$ and $1$ inclusive.
- Random variable independent of itself
- Difference between “space” and “algebraic structure”