Open Intervals are Not Compact

Topology · Axiom Academy

EXAMPLE Open Intervals are Not Compact Three ways to prove (0,1) has no finite subcover Using Nested Intervals Near Zero Using Expanding Intervals from Center Using the Complement of a Converging Sequence Open intervals in are never compact because they lack their boundary points. All three examples exploit the same fundamental issue: we can construct open covers that "approach" the missing endpoints but never quite reach them. Example 1 uses intervals that shrink toward 0, Example 2 uses intervals expanding from the center, and Example 3 uses complements of sequence points. The Heine-Borel theorem tells us that in , a set is compact if and only if it's closed and bounded. (0,1) is bounded but not closed. Understanding these constructions helps build intuition for why compactness requires both closedness and boundedness in Euclidean spaces.

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