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Motivating the terminology of weaker vs stronger topology
Let $X$ be a set, $\tau_1,\tau_2$ two topologies on $X$, and consider the following statements$\tau_1\subseteq \tau_2$ (i.e $\tau_1$ is coarser/weaker than $\tau_2$, or that $\tau_2$ is finer/stronger...
View ArticleAnswer by Saúl RM for Motivating the terminology of weaker vs stronger topology
The converse is true: let´s prove $\neg1\implies\neg2$. If you have a set $U$ in $\tau_1$ but not in $\tau_2$, you can pick a point $x\in U$ which isn´t in the interior of $U$ respect to the topology...
View ArticleAnswer by Henno Brandsma for Motivating the terminology of weaker vs stronger...
Suppose (2) holds. Let $O \in \tau_1$. Then $X\setminus O$ is closed in $\tau_1$, and it's also closed in $\tau_2$: let $x$ be in the closure (w.r.t $\tau_2$) of $X\setminus O$, so there is a net...
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