differential-geometry
Theorem. Every Manifold is locally compact.
This is a problem in Spivak's Differential Geometry.
However, don't know how to prove it. It gives no hints and I don't know if there is so stupidly easy way or it's really complex.
I good example is the fact that Heine Borel Theorem, I would have no clue on how to prove it if I didn't see the proof.
So can someone give me hints. I suppose if it's local, then does this imply that it's homeomorphic to some bounded subset of a Euclidean Space?
A couple points:
[2016-07-25]: Section Differential Geometry added.
Although OP narrowed down the post, there are still many more important historical facts which should be addressed to adequately answer the question, than I can give in this answer. Nevertheless here are some aspects, which might be interesting.
At least we will see, OP is right when he thinks that many different candidate definitions of manifolds competed to become the most suitable one.
We start with question (5), good books addressing the history of differential geometry/topology.
A History of Algebraic and Differential Topology 1900 - 1960 by Jean Dieudonné.
I strongly recommend this book which provides a wealth of historical information as well as technical details. Most of it is regrettably beyond my scope, but it's great for me to get at least a glimpse of the exciting development when going through some parts of the book.
In what follows I focus on OPs question (1) and provide some small samples of text mostly cited verbatim from the book.
As we can read in chapter I, the modern development started with the work of Poincaré. It was his groundbreaking long paper Analysis Situs published in 1895 and followed by five so-called Complements between 1899 and 1905.
The Work of Poincaré
[ch 1, § 1.] Concepts and results belonging to algebraic and differential topology may already be noted in the eighteenth and nineteenth centuries, but cannot be said to constitute a mathematical discipline in the usual sense.
Before Poincaré we should therefore only speak of the prehistory of algebraic topology; ...
But note that topological space has not yet been defined at that time. But some intuitive notion of manifolds was already available.
Of course, before Frèchet (1906) and Hausdorff (1914) the general notion of topological space had not been defined;
what had become familiar after the work of Weierstrass and Cantor were the elementary topological notions (open sets, closed sets, neighborhoods, continuous mappings, etc.) in the spaces $\mathbb{R}^n$ and their subspaces;
these notions had been extended by Riemann (in an intuitive way and without any precise definition) to $n$-dimensional manifolds (or rather what we now would call $C^r$-manifolds with $r\geq 1$).
In this chapter I Dieudonné examines rather detailed Poincaré's Analysis Situs. He explains that Poincaré was the first who introduced the idea of computing with topologial objects, not only with numbers. Most important he introduced the concepts of homology and fundamental group.
With respect to manifolds we also find:
Poincaré appealed to the concept of oriented manifold by Klein for surfaces and generalized by von Dyck for manifolds of arbitrary dimension. In Analysis Situs, Poincaré gave a characterization of orientable manifolds by what is still one of the modern criteria:
there exist charts $(U_\lambda,\psi_{\lambda})$ such that the transitition diffeomorphisms $$\psi_{\lambda}\left(U_\lambda\cap U_{\mu}\right)\rightarrow\psi_{\mu}\left(U_{\lambda}\cap U_{\mu}\right)$$ have positive determinants for all pairs of indices such that $U_{\lambda}\cap U_{\mu}\neq \emptyset$.
But later on Dieudonné addresses also some weak points in connection with this definition. In fact we won't find some final definition of the term manifold by Poincaré as we can read in the next paragraph. Nevertheless the far-reaching character of his work is tremendous:
... As in so many of his papers, he gave free rein to his imaginative powers and his extraordinary intuition, which only very seldom led him astray; in almost every section is an original idea.
But we should not look for precise definitions, and it is often necessary to guess what he had in mind by interpreting the context. ...
Thus ends this fascinating and exasperating paper, which, in spite of its shortcomings, contains the germs of most of the developments of homology during the next 30 years.
In section I. § 4 Duality and Intersection Theory on Manifolds there is a subsection $A$ titled with
The Notion of Manifold
[ch 1, § 4.A] After the invariance problem had been solved, two main items remained in the implementation of the program outlined by Poincaré: a rigorous proof of the duality theorem and a complete theory of intersection barely begun by Poincaré.
Obvious examples show that in neither question can one work with a general cell complex; some restrictions have to be introduced in order to make available the arguments Poincaré used for his manifolds.
...
In the meantime, in order to use simplicial methods, topologists had to settle for more tractable definitions of manifolds. In fact, several definitions were proposed ([308], pp. 342-343);
Dieudonné refers here to Algebraic Topology by S. Lefschetz from 1942. We can find there $9$ different types of manifolds, all of which are supposed to be $n$-dimensional.
Combinatorial manifolds: Let $X=Y-Z$ be an open simple complex, where $Y$ is closed simple and $Z$ is a closed subcomplex of $Y$. We say that $X$ is an orientable combinatorial manifold whenever the following two conditions are fulfilled.
(1) The dual $X^\star$ of $X$ has a closed simple weak isomorph $\overline{X}$.
(2) If $x,x^\prime$ are distinct elements of $X$, then $\text{Cl } x \cap \text{St } x^\prime$ is acyclic or void.
They may be finite or infinite, absolute or relative, orientable or non-orientable, simplicial or merely simple complexes.
Geometric manifolds: Euclidean realizations of the preceding simplicial types.
Manifolds in the sense of Brouwer: Euclidean complexes such that the star of each vertex is isomorphic with a set of simplexes in an Euclidean $\mathcal{E}^n$ having a common vertex $P$ and making up a neighborhood of $P\in\mathcal{E}^n$.
Manifolds in the sense of Newman: Euclidean complexes such that if $a$ is a vertex $\text{St} a = aB$, then $B$ is partition-equivalent to an $(n-1)$-sphere.
Manifolds in the sense of Poincaré and Veblen: Topological complexes such that every point has for neighborhood an $n$-cell.
Topological manifolds: An $M^n$ of this type is separable metric space with a countable locally finite open covering consisting of $n$-cells. Noteworthy special cases: $C^r$-manifolds, differentiable manifolds, analytical manifolds, $\Gamma$-manifolds, group manifolds.
Generalized manifolds: Locally compact spaces discussed by Čech, Lefschetz, Wilder and others and characterized by certain properties of so-called local connectedness or local connectedness in the sense of homology and also by the property: each point is $n$-cyclic.
Pseudo-manifolds: This term has been applied by Brouwer and other authors to what we have called a simple geometric $n$-circuit.
Manifolds of grade $p$: Simplicial $n$-complexes, investigated by Čech, and which behave like an $M^n$ only as regards the two consecutive dimensions $p-1,p$.
Of course this answer hardly touches the surface of information provided in J. Dieudonné's book. ... curious? :-)
A nice historical survey is The Concept of Manifold, 1850-1950 by Erhard Scholz. Section 5 is devoted to the development of the modern manifold concept. In subsection 5.4 he describes the birth of the "modern" axiomatic concept in differential geometry.
Differential Geometry
There was, of course still another line of research, more closely linked to differential geometry, where manifolds played an essential role, and purely topological aspects (independently of whether continuous, combinatorial, or homological ones) did not suffice and still needed elaboration. ...
and later on
Whitehead and Veblen presented their axiomatic characterization of manifold of class $G$ first in a research article in the Annals of Mathematics (1931) and in the final form in their tract on the Foundations of Differential Geometry (1932).
Their book contributed effectively to a conceptual standardization of modern differential geometry, including not only the basic concepts of continuous and differentiable manifold of different classes, but also the modern reconstruction of the differentials $dx=(dx_1,\ldots,dx_n)$ as objects on tangent spaces to $M$.
Basic concepts like Riemannian metric, affine connection, holonomy group, covering manifolds, etc. followed in a formal and symbolic precision that even from the strict logical standards of the 1930-s there remained no doubt about the wellfoundedness of differential geometry in manifolds.
Best Answer
By definition, if $X$ is a manifold, then every point $x \in X$ admits an open neighborhood $U$ which is homeomorphic to $\mathbb{R}^n$ ($n$ is allowed to depend on $x$). Let $f: U \rightarrow \mathbb{R}^n$ be such a homeomorphism. Let $B$ be a closed ball of finite radius about $f(x)$ in $\mathbb{R}^n$. By Heine-Borel, $B$ is compact, hence so is its homeomorphic preimage $f^{-1}(B)$, which is therefore a compact neighborhood of $x$.
Almost the same argument shows that $X$ has even a neighborhood base of compact sets at every point, which for non-Hausdorff spaces, is a priori stronger than having a single compact neighborhood at any point. In my opinion "locally compact" should mean this stronger condition. (On the other hand, in my terminology, both "manifold" and "locally compact" include the Hausdorff condition.)