Define $X^{\{n\}}:=\{A\subseteq X:|A|=n\}$, the set of subsets of cardinality $n$. If $X$ is a topological space, $X^{\{n\}}$ can be given a topology by considering it to be a quotient of $X^n$ minus the extended diagonal. Define a continuous $n$-partition of a space $X$ to be a partition of $X$ ...

Let $X$ be a manifold and suppose $f:X\to X^{\{n\}}$ is a continuous $n$-partition. Let $p:X\to Y$ be the quotient map associated to this partition of $X$. I claim that in fact $p$ is a covering map. (Conversely, it is easy to see that any partition that comes from an $n$-sheeted covering map ...

I am reading E-Li-Vanden-Eijnden's Applied Stochastic Analysis, and the author claims: The recurrence relation of $\boldsymbol{\mu}_n=\boldsymbol{\mu}_{n-1}\boldsymbol{P}$, where $\boldsymbol{\mu}_n$ is the distribution of $X_n$, and $\boldsymbol{P}$ is the transition matrix, can be written as $$...

Let $X$ be a cell complex that is a union of two contractible subcomplex whose intersection is contractible. Then $X$ is contractible. I found some solution: Let $X = U\cup V$ with $W = U\cap V$, where $U,V,W$ are all contractible subcomplexes. First consider the quotient map $q:X\to X/W=:X_1$ ...