Subject GRE-Style: Topological Properties

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Topological Properties 3 worked problems on connectedness, compactness, Hausdorff spaces, and homeomorphisms Key Definitions: A space is connected if it cannot be written as a union of two disjoint nonempty open sets. A space is Hausdorff if distinct points have disjoint open neighborhoods. Two spaces are homeomorphic if there is a continuous bijection with continuous inverse. Question: Which of the following subsets of is connected? Key Fact: A subset of is connected if and only if it is an interval (possibly unbounded, open, closed, or half-open). (A) : Not an interval. For example, but . is totally disconnected. Not connected. (B) : Discrete set, not an interval. Not connected. (C) : Gap between 1 and 2. Not an interval. Not connected. (D) : This equals , which is an interval. Connected! (E) : Countable set with gaps. Not an interval. Not connected. GRE Tip: Before checking open-cover definitions, first simplify the set. Many "union" expressions collapse to intervals. Question: Which of the following pairs of spaces are homeomorphic? (A) vs. : is compact, is not. Compactness is a topological invariant, so they are not homeomorphic. (B) vs. : Both are open, connected, non-compact. The map is a homeomorphism from to . Homeomorphic! (C) vs. : Removing any point from leaves it connected, but removing an interior point from disconnects it. Not homeomorphic.

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