Super-simplified: What is a topology

‘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.

* ‘synthesized’ in here means constructed from union and/or finite intersections.

[Bonus] What I learned from real line topology in real analysis 101:

  1. Normal intuitive cases
  2. Null and universal set are clopen
  3. Look into rationals (countably infinite) and irrationals (uncountable)
  4. Blame Cantor (sets)!



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