m5-normed-spaces

en m3-set-theory definimos en basiquísimo concepto de set, y en el capitulo anterior vimos cómo agregarle the structure necesaria para realizar operaciones algebraicas con sus miembros. En esta entrega nos centraremos en las estructuras que nos permiten hablar formalmente del concepto de distancia entre sus miembros.

Empezaremos por definir el concepto de distancia más básico.

topology

Given a set $X$, a topology is a set $\mathcal{O}$ of subsets of $X$ that contains $\varnothing$ and $X$, and is closed-under unions and intersections. Any set $O\in \mathcal{O}$ is called open, and any subset $C\subseteq X$ is called closed if its complement is open. A set that is both open and closed, is called clopen (e.g. $\varnothing$ and $M$). If we have two topologies ${\mathcal{O}}_1$ and ${\mathcal{O}}_2$ s.t. ${\mathcal{O}}_1\subseteq {\mathcal{O}}_2$ we say that ${\mathcal{O}}_2$ is a finer topology than ${\mathcal{O}}_1$ and that ${\mathcal{O}}_1$ is coarser than ${\mathcal{O}}_2$, and if ${\mathcal{O}}_1$ is a proper-subset (i.e. ${\mathcal{O}}_1\subseteq {\mathcal{O}}_2$) we add the word “strict”, i.e., we say that ${\mathcal{O}}_2$ and ${\mathcal{O}}_1$ are, respectively, a strict-finer and a strict-coarser topology than the other.

Examples

Note: Number of topologies given the cardinality of $X$.

Note: $\subseteq$ defines a partial-order over the topologies of $X$. todo ver si agrego y explico esto acá o más adelante.

metric

norm