Topology
What is preserved when distance is forgotten?
Take away distance and keep only nearness. Almost everything you expected to survive does not, a few things unexpectedly do, and those few turn out to be the reason analysis works at all.
Open sets
What is the least you need to say what near means?
Nearness without a ruler
Continuity was defined with epsilon and delta, both of which are distances. The film removes them one at a time and finds what continuity actually needed.
The same space, two different topologies
One set can carry many topologies, and the choice changes which functions are continuous. The film makes that concrete on a set of three points.
Compactness
What replaces finiteness in an infinite space?
Every cover has a finite subcover
The definition is notoriously unintuitive. The film arrives at it by asking which theorems about closed intervals actually used closedness and boundedness.
Why compactness is the useful hypothesis
Continuous functions on compact sets attain their maxima, are uniformly continuous, and behave. One property, three theorems.
Connectedness
When is a space genuinely in one piece?
Cutting a space without cutting anything
Connectedness is defined by the impossibility of a certain kind of split, which is a strange way to define wholeness until you try the alternatives.
The intermediate value theorem, again
The theorem proved in Analysis using completeness turns out to be a statement about connectedness, and the topological proof is three lines.
The fundamental group
Can a loop tell you the shape of the space it lives in?
Topology, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume X ↓