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Articles by Lee Mosher
section of a fiber bundle
Let $M$ be a Möbius strip, $C\subset M$ a circle running through the center, and consider the fiber bundle $M\setminus C \to C$ given by some projection. What happens if we try to construct a section in the way you describe? For each $c\in C$, we need to pick a point either above or below $C$. But as you can easily verify with some paper, $M\setminus C$ is connected! So if we "follow it through", we will find that there is no consistent choice of side.
analysis - "Vectors aren't really numbers" - how sound is that statement? - Mathematics Stack Exchange
Since I first learned about vectors, I noticed something interesting: almost any numeric formula can be replaced by a vectorial formula by just replacing addition, multiplication, etc., with their vectorial elementwise versions. For example: not only gets the average of 2 numbers, but the mid-point between 2 vectors if you use the elementwise addition. On Calculus, many formulas for integrals can be extended to triple, quadruple integrals unchanged the same way.
Confusion about concept of basis in point set topology.
First off, here is the formal definition of being a basis for a topology (notice that I bolded the word "a" -- after you read my answer, if you don't understand why, please comment): Let $X$ be a set. Let $\mathcal{A}$ be a collection of subsets of $X$. Then $\mathcal{A}$ forms a basis for a topology if: Notice that nowhere in the definition above did we say that $X$ is already a topological space with a given topology. We just said $X$ is a set.
Why is the slope of a line defined the way it is.
Probably the formula is the way it is for historical reasons. Generally speaking, our intuition of "steepness" matches the definition of slope: the steeper a line is, the higher is its slope in absolute value. This works with slope equal to $\text{rise} / \text{run}$, but would not work if the slope were equal to $\text{run} / \text{rise}$ as you suggest. For a modern example, when you drive down a steep mountain highway, you will notice signs saying something like "Danger: 5% grade ahead.
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