Intereting Posts

Block inverse of symmetric matrices
Importance of Constructible functions
Applications of equidistribution
How many subspace topologies of $\mathbb{R}$?
Why are the domains for $\ln x^2$ and $2\ln x$ different?
When is $991n^2 +1$ a perfect square?
Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$
Geometrically Integrating $R(x,\sqrt{Ax^2+Bx+C})$ & Motivating Euler Substitutions
Limit of integral without Taylor expansion
Why do they use $\equiv$ here?
How do I find upper triangular form of a given 3 by 3 matrix??
If every convergent subsequence converges to $a$, then so does the original bounded sequence (Abbott p 58 q2.5.4 and q2.5.3b)
Product of two probability kernel is a probability kernel?
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
Bounded index of nilpotency

Question:a Hausdorffdifferentiablemanifold (locally Euclidean space): $$ \text{is metrizable} \iff \text{is paracompact} \iff \text{admits a Riemannian metric} \,?$$Does one also have for a locally Euclidean Hausdorff space (not necessarily differentiable): $$\text{second countable} \iff \text{metrizable with countably many connected components}? $$

Thus second countability is

strictlystronger for such spaces than metrizability/paracompactness/existence of a Riemannian metric?

(According to this question, it seems they are all equivalent when connectedness is assumed, and according to this question even equivalent to the existence of a universal cover when connected.)

**Attempt:** By the Smirnov metrization theorem, a locally metrizable space (e.g. locally Euclidean spaces) are metrizable if and only if they are Hausdorff and paracompact.

According to Wikipedia, Riemannian manifolds are metrizable.

(Is this only true for connected Riemannian manifolds, or can we use the trick where we make the metric less than 1 on each connected component and 1 for distances between points in different components? Or perhaps only when there are at most countably many components?)

Finally, according to Wikipedia, any ** differentiable** paracompact manifold admits a Riemannian metric (I’m not sure if the differentiable hypothesis is necessary — this question seems related).

Thus (at least for *connected*, differentiable Hausdorff manifolds) “admits Riemannian metric” $\implies$ “metrizable” $\implies$ paracompact $\implies$ “admits Riemannian metric”.

**Context:** This question is motivated by how the definition of manifold used by Spivak in *Comprehensive Introduction to Differential Geometry* (see here for a related question) is different from the one used by Lee in *Introduction to Smooth Manifolds* (where second countability is required). In particular, I had thought that the definition my professor was using (Hausdorff, metrizable, locally Euclidean) was equivalent to Lee’s, until Spivak started mentioning a bunch of counterexamples which I recognized as not being second countable. Although I don’t remember if my professor specified countably many components, which would rule out most of Spivak’s counterexamples consisting of uncountable spaces with the discrete topology and metric.

- Geodesic equations and christoffel symbols
- What is an intuition behind total differential in two variables function?
- The function that draws a figure eight
- Liouville form on the cotangent bundle
- Geodesics on a polyhedron
- The degree of antipodal map.
- Why is the Riemann curvature tensor the technical expression of curvature?
- Can $S^4$ be the cotangent bundle of a manifold?
- Volume form on $(n-1)$-sphere $S^{n-1}$
- What do we mean when we say a differential form “descends to the quotient”?

It seems like the answer to many of these questions can be found on p.459, Appendix A, of Spivak’s *Comprehensive Introduction to Differential Geometry*, 3rd edition 1999.

Specifically, the Theorem appaears to address my confusion regarding how the cardinality of the number of components of a manifold plays into these properties/definitions.

Note that this seems to hold even for topological manifolds, i.e. not necessarily smooth ones.

TheoremThe following properties are equivalent for any manifold $M$:(a) Each component of $M$ is $\sigma$-compact.

(b) Each component of $M$ is second countable (has a countable base for the topology).

(c) $M$ is metrizable.

(d) $M$ is paracompact.(In particular, a compact manifold is metrizable.)

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- Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p$
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- Computing an awful integral
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- Counting number of solutions with restrictions
- Prove that $2^n < \binom{2n}{n} < 2^{2n}$
- Why is “$P \Rightarrow Q$” equivalent to “$\neg Q \Rightarrow \neg P$”?