Which topological property does this argument actually need?
Open, closed, compact, and simply connected are four separate properties -- mixing them up breaks a proof.
Each property answers a different question. OPEN asks "does every point have room around it?" CLOSED asks "does the set contain its own edge?" COMPACT (closed AND bounded) is what the Extreme Value Theorem needs to guarantee an absolute max/min exists at all. SIMPLY CONNECTED is a completely different axis -- it's about holes, not size, and it's what decides whether a closed form is automatically exact.
Room around a point, versus containing the edge
Term
Open set
A set where every point has some wiggle room -- a small enough ball around it still fits inside.
Term
Closed set
A set that contains all of its own boundary points.
A point where every ball, no matter how small, catches points both inside and outside the set.
Guaranteeing an absolute extremum exists
Term
Compact set
Closed AND bounded -- the property that guarantees an absolute max and min exist.
Holes, not size
A region with no holes -- every loop drawn inside it can be shrunk down to a point without leaving the region.
The measurement of exactly how many closed forms on a region fail to be exact -- a number that detects holes.
A region failing to be simply connected is exactly what lets a closed form fail to be exact.
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