Precompact dots

Let M, N be metric spaces such that M is connected and N is complete and locally compact. Let \mathcal{F} be a uniformly equicontinuous family of functions from M to N such that there exists x_0\in M with the property (\star) that \mathcal{F}(x_0) is precompact. Prove that all points x\in M have property (\star).

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Precompact dots

Separating the étalé space

We briefly recall the construction of the étalé space of a sheaf. Given a sheaf F over a space X, the associated étalé space is a topological space E which, as a set, is the disjoint union of the stalks of F, with the topology induced by the basis B(s, U) = \{s_x : x\in U\}, where U is an open set in X, s\in F(U) and s_x is the germ at x of the element s.

Given a topological space X, consider the sheaf of real-valued continuous functions on X, which we denote C. It is more or less easy to see that the associated étalé space is not Hausdorff iff the space X has the following property (\star):

There exist continuous functions f,g:U\to\mathbb{R}, where U is an open set, and a point x\in U such that the germs of f and g at x are different, but nevertheless for any open neighborhood V\subseteq U of x there exists an open set W\subseteq V such that f|_W = g|_W.

For instance, \mathbb{R} has property (\star), and it’s easy to see that (\star) is inherited by subspaces and products which contain any (\star) factor. This, for instance, shows that subspaces of \mathbb{R}^n are spaces satisfying this property.

Is there any characterization of all topological spaces for which the étalé space associated with the sheaf of real-valued continuous functions is not Hausdorff?

Separating the étalé space

Your degree is a chance for expansion

Let X and Y be compact oriented smooth manifolds of the same dimension, with Y connected. Let f : X \to Y be a smooth function. Take y \in Y a regular value of f and, for x \in f^{-1}(y), define o(x) as +1 if f preserves orientation at x and as -1 if it reverses the orientation. We define the degree of f as the number d(f) = \sum_{x\in f^{-1}(y)}o(x).

Prove the following extension theorem: Assume Y=S^k is a k-dimensional sphere and X=\partial W is the boundary of a compact oriented (k+1)-manifold W. There exists a smooth extension F:W\to Y of f if and only if d(f)=0.

Your degree is a chance for expansion