Let $(X,x_0),(Y,y_0)$ be two pointed topological spaces. Then $X\vee Y$ obtained by $X\amalg Y/x_0\sim y_0$ where $X\amalg Y$ denote the disjoint union of topological spaces and $X\times\{y_0\}\cup \{x_0\}\times Y\subset X\times Y$: subspace are homeomorphic. My plan to prove this is first by u...

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k...

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$. (a) Is it true, that the manifold B can be covered by a finite number of sets $U_1,\dots,U_N$ s.t. the v...