Compactness

• Is There a Compact Set That Is Not Jordan Measurable?
  Yes. The Smith–Volterra–Cantor set is compact, has positive measure, and has empty interior, so its boundary is not null and it is not Jordan measurable.
• When Is a Proper Injective Map a Homeomorphism onto Its Image?
  A continuous proper injective map into a first-countable Hausdorff space is an embedding; without Hausdorffness, the statement can fail.
• What Is the Relationship Between Compact and Hausdorff Spaces?
  Compactness and the Hausdorff property are independent, but together they imply useful results: compact subsets are closed and certain bijections are homeomorphisms.
• Does the Bolzano–Weierstrass Property Imply Countable Compactness?
  Yes: sequential compactness implies countable compactness, and the limit-point form of Bolzano–Weierstrass does too for T1 spaces.
• A Simple Example of a Lindelöf Space
  The real line is a simple Lindelöf space: its countable rational-interval basis lets you extract a countable subcover from any open cover.
• Is Every Subset of a Compact Set Compact?
  A subset of a compact space is not necessarily compact. Closed subsets are compact, and in compact Hausdorff spaces, compact subsets are exactly the closed subsets.