Hatcher's construction of a CW complex is as follows: (1) Start with a discrete set $X^0$ , whose points are regarded as $0$ cells. (2) Inductively, form the $n$ skeleton $X^n$ from $X^{n−1}$ by attaching $n$ cells $e_{\alpha}^n$ via maps $\phi_{\alpha}:S^{n-1} \to X^{n-1}$. This means that $X^n$...

My first question/confusion is regarding the definition of reduced suspension $\sum X$ and suspension $SX$ of a space $X$. Usually, we denote the quotient space of $X$ with subspace $A$ identified to a point as $X/A$. I understand that suspension of a space is like a 'double-cone': we take a spac...

In Loring Tu's book introduction to manifolds, the definition of a local flow, page 223-224, the following is said: For a smooth vector field $X$ on $M$ and $p$ in $M$, there exists an open set $U$ of $p$ and an $\epsilon>0$ and a smooth map $\varphi: (-\epsilon,\epsilon)\times U \rightarrow M$ s...

Are fields $Q[i\sqrt[4]{5}]$ and $Q[\sqrt[4]{5}]$ isomorphic? I tried to prove they are not simiralry like one can show that $Q[i\sqrt{5}]$ and $Q[\sqrt{5}]$ are not isomorphic because if they were then $(f(i\sqrt{5})^2=f(-5)=-5$ but in $Q[\sqrt{5}]$ there is no element such that it's square is n...