Limit Point Compactness

Topology · Axiom Academy

LESSON Limit Point Compactness Understanding another notion of compactness in topological spaces Step 1: What is Limit Point Compactness? We've seen compactness defined through open covers. Now we explore an alternative characterization that focuses on infinite subsets and their accumulation points. A topological space is limit point compact (also called weakly countably compact ) if every infinite subset of has a limit point in . In other words, you cannot have an infinite set that is completely "spread out" with no accumulation point. Every infinite collection of points must cluster somewhere in the space. Step 2: Revisiting Limit Points Let's recall what it means for a point to be a limit point of a set. A point is a limit point of a set if every neighborhood of contains a point of distinct from . In with the standard topology, consider the sequence . The point is a limit point of this infinite set because every neighborhood of 0 contains infinitely many points from the sequence. Step 3: Examples and Non-Examples Let's build intuition by examining specific spaces. [0, 1] with standard topology: Any infinite subset has a limit point by Bolzano-Weierstrass Any finite space: Vacuously true (no infinite subsets) Cofinite topology on infinite set: Complement of any point is the only large open set Discrete space (infinite): Any infinite subset has no limit point (every point is isolated) ℝ with standard topology: The integers ℤ form an infinite set with no limit point

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