General topology

• 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.
• Can a Closed Set Have No Nonempty Open Subsets?
  Yes. The Cantor set is closed in the real line but contains no nonempty open interval; this means its ambient interior is empty.
• 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.
• What Are the Analytic Applications of the Stone–Čech Compactification?
  The Stone–Čech compactification turns bounded continuous functions on X into continuous functions on a compact space, aiding analysis and functional analysis.
• 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.
• Are Singleton Sets in ℝ Both Open and Closed?
  In ℝ with the usual topology, a singleton {a} is closed but not open. Proofs of both facts, and why only ∅ and ℝ are both open and closed.
• Is Every Singleton Set Open? (No, Here's When It Is)
  Singleton sets are not open in general. In ℝ, {0} is closed but not open. A singleton {x} is open exactly when x is an isolated point, as in discrete spaces.
• Why Is Every Singleton Set Open in a Finite Metric Space?
  Proof that {x} is open in any finite metric space, using the minimum distance to the other points, and why this makes every subset open (the discrete topology).