Super-simplified: What is a topology | Rambling Nerd with a Plan

‘Super-simplified’ is my series of brief notes that summarizes what I have learned so I can pick it up at no time. That means summarizing an hour of lecture into a few takeaway points.

These lectures complemented my gap in understanding open sets in undergrad real analysis, which I understood it under the narrow world-view of the real line.

$X$ : Universal set

Topology ≡ open + $\left\{\varnothing, X\right\}$

Open ≡ preserved under unions, and finite intersections.

Why finite needed for intersections only? Infinite intersections can squeeze open edge points to limit points, e.g. $\bigcap^{\infty}_{n}(-\frac{1}{n},\frac{1}{n}) = \left\{0\right\}$ .

Never forget that $\left\{\varnothing, X\right\}$ is always there because it might not have properties that the meat open set $B$ doesn’t have. e.g. a discrete topology of $\mathbb{Q}$ on $(0,1) = B$ $\subseteq$ universal set $X=\mathbb{R}$ means for any irrational point, $\mathbb{R}$ is the only open-neighborhood (despite it looks far away) because they cannot be ‘synthesized*’ from $\mathbb{Q}$ using operation that preserves openness.