Topology

This small note is an introduction to Topology that follows the introductory arguments of (Armstrong 2013). Euler’s Theorem We will start our journey in topology following a classical example in the history of Mathematics the relation: $$ v - e + f = 2 $$ Valid for classical Polyhedrons. Basic definitions Polyhedron It’s a collection of plane polygons (see Programmazione lineare#Poliedro) such that: Every polygon shares each of its edges with exactly another polygon We have vertexes that can be shared by many polygons....

Introduction to topological spaces We want now to extend the idea of continuity presented in limits, which is a function $f : E^{n} \to E^{n}$ is continuous if given $x$ then $\forall\varepsilon > 0$ $\exists \delta$ such that $\forall y : \lVert y -x \rVert < \delta \implies \lVert f(y) - f(x) \rVert < \varepsilon$. But we want to get rid of the idea of distance, and base our definition on the idea of neighborhoods, which in $E^{n}$ are just spherical radius centered around a point....

There is a close relationship between topologies and metric spaces. We will see that every metric space directly induces a topology based on its metric. (from a CS point of view, this means topologies are more general than metric spaces). Definition of Metric Space 🟩 We say that $(\mathcal{X}, d)$ is a metric space if $\mathcal{X}$ is a set and $d$ a function $\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ such that:...