The Math Graduate

Q. Show that $(\mathbb{Z}/m\mathbb{Z})\otimes (\mathbb{Z}/n\mathbb{Z}) = 0$ if $m$ , $n$ are coprime.

A. $m$ , $n$ are coprime. Hence, there are integers $a, b$ such that $am+bn=1$ . Consider an element $\bar{k}\otimes \bar{l}$ , a typical element of the generating set of $(\mathbb{Z}/m\mathbb{Z})\otimes (\mathbb{Z}/n\mathbb{Z})$ . We have $\bar{k}\otimes \bar{l} = (am+bn)(\bar{k}\otimes \bar{l}) = (am)(\bar{k}\otimes \bar{l}) + (bn)(\bar{k}\otimes \bar{l}) = a(\bar{mk}\otimes \bar{l}) + b(\bar{k}\otimes \bar{nl}) = 0$ .

Munkres

Munkres – Section 23 Problem 1

October 5, 2017October 5, 2017 JDF

Q. Let $\mathcal{T}$ and $\mathcal{T}'$ be topologies on $X$ . If $\mathcal{T}' \supseteq \mathcal{T}$ , what does connectedness of $X$ in topology imply about connectedness in the other?

A. Let $(X,\mathcal{T}')$ be connected. Then there is no separation of $X$ in $\mathcal{T}'$ . Hence, there is no separation of $X$ in $\mathcal{T}$ since there are no new open sets. So, $(X,\mathcal{T})$ is connected.

Let $(X,\mathcal{T})$ be connected. We cannot say anything about connectedness in $\mathcal{T}'$ , as there are more open sets in $\mathcal{T}'$ than $\mathcal{T}$ .