‘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.
: Universal set
Topology ≡ open +
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. .
Never forget that is always there because it might not have properties that the meat open set doesn’t have. e.g. a discrete topology of on universal set means for any irrational point, is the only open-neighborhood (despite it looks far away) because they cannot be ‘synthesized*’ from 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:
- Normal intuitive cases
- Null and universal set are clopen
- Look into rationals (countably infinite) and irrationals (uncountable)
- Blame Cantor (sets)!