Countable Metric Spaces Without Isolated Points Frederick K. Dashiell Jr. CECAT - Center of Excellence for Computation, Algebra, and Topology Chapman University, Orange, California and University of California at Los Angeles - UCLA [email protected] June 11, 2021 I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) The Challenge To prove directly: I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) The Challenge To prove directly: Q × Q ≈ Q I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! A standard modern proof (Engelking 1995: exercises with hints) The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I X ≈ Q I Not direct! The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I Not direct! The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q The Challenge To prove directly: Q × Q ≈ Q Q \ (0;1] ≈ Q \ (0;1) A standard modern proof (Engelking 1995: exercises with hints) I Any countable metric space is homeomorphic to a subspace of Cantor set ∆. I (Brouwer 1910) Every zero-dimensional compact metric space with no isolated points is homeomorphic to ∆. I If X is a countable subset of ∆ with no isolated point, then X ≈ ∆. I Any two countable dense subsets of ∆ are homeomorphic. I X ≈ Q I Not direct! I 1921 Any countable metric space is homeomorphic to a subspace of R – proof 5 pages I 1934 Introduction to General Topology — states general case but proof is incomplete, no citation I 1956 General Topology (reprint Dover 2000) — proof incomplete, no citation I Textbook proofs are rare! Sierpinski´ I 1920 Any two countable dense-in-itself subsets of Rn are homeomorphic – proof 6 pages I 1934 Introduction to General Topology — states general case but proof is incomplete, no citation I 1956 General Topology (reprint Dover 2000) — proof incomplete, no citation I Textbook proofs are rare! Sierpinski´ I 1920 Any two countable dense-in-itself subsets of Rn are homeomorphic – proof 6 pages I 1921 Any countable metric space is homeomorphic to a subspace of R – proof 5 pages I 1956 General Topology (reprint Dover 2000) — proof incomplete, no citation I Textbook proofs are rare! Sierpinski´ I 1920 Any two countable dense-in-itself subsets of Rn are homeomorphic – proof 6 pages I 1921 Any countable metric space is homeomorphic to a subspace of R – proof 5 pages I 1934 Introduction to General Topology — states general case but proof is incomplete, no citation I Textbook proofs are rare! Sierpinski´ I 1920 Any two countable dense-in-itself subsets of Rn are homeomorphic – proof 6 pages I 1921 Any countable metric space is homeomorphic to a subspace of R – proof 5 pages I 1934 Introduction to General Topology — states general case but proof is incomplete, no citation I 1956 General Topology (reprint Dover 2000) — proof incomplete, no citation Sierpinski´ I 1920 Any two countable dense-in-itself subsets of Rn are homeomorphic – proof 6 pages I 1921 Any countable metric space is homeomorphic to a subspace of R – proof 5 pages I 1934 Introduction to General Topology — states general case but proof is incomplete, no citation I 1956 General Topology (reprint Dover 2000) — proof incomplete, no citation I Textbook proofs are rare! I D = fdX (xm;xn): m 6= ng [ fdY (ym;yn): m 6= ng I D ⊂ (0;¥) countable I S ⊂ X;Y : next(S) = xk where k = minfi : xi 2 Sg I ball: B(x;r) = fy : d(x;y) < rg; r > 0 I open set: union of balls I isolated: fxg is a ball I continuous function: e;d I homeomorphism: bijection, continuous in both directions Theorem Metric X = fx1;x2;:::g,Y = fy1;y2;:::g no isolated points =) X ≈ Y Proof: Elementary proof with finite recursion I Toolbox I D = fdX (xm;xn): m 6= ng [ fdY (ym;yn): m 6= ng I D ⊂ (0;¥) countable I S ⊂ X;Y : next(S) = xk where k = minfi : xi 2 Sg I open set: union of balls I isolated: fxg is a ball I continuous function: e;d I homeomorphism: bijection, continuous in both directions Theorem Metric X = fx1;x2;:::g,Y = fy1;y2;:::g no isolated points =) X ≈ Y Proof: Elementary proof with finite recursion I Toolbox I ball: B(x;r) = fy : d(x;y) < rg; r > 0 I D = fdX (xm;xn): m 6= ng [ fdY (ym;yn): m 6= ng I D ⊂ (0;¥) countable I S ⊂ X;Y : next(S) = xk where k = minfi : xi 2 Sg I isolated: fxg is a ball I continuous function: e;d I homeomorphism: bijection, continuous in both directions Theorem Metric X = fx1;x2;:::g,Y = fy1;y2;:::g no isolated points =) X ≈ Y Proof: Elementary proof with finite recursion I Toolbox I ball: B(x;r) = fy : d(x;y) < rg; r > 0 I open set: union of balls I D = fdX (xm;xn): m 6= ng [ fdY (ym;yn): m 6= ng I D ⊂ (0;¥) countable I S ⊂ X;Y : next(S) = xk where k = minfi : xi 2 Sg I continuous function: e;d I homeomorphism: bijection, continuous in both directions Theorem Metric X = fx1;x2;:::g,Y = fy1;y2;:::g no isolated points =) X ≈ Y Proof: Elementary proof with finite recursion I Toolbox I ball: B(x;r) = fy : d(x;y) < rg; r > 0 I open set: union of balls I isolated: fxg is a ball I D = fdX (xm;xn): m 6= ng [ fdY (ym;yn): m 6= ng I D ⊂ (0;¥) countable I S ⊂ X;Y : next(S) = xk where k = minfi : xi 2 Sg I homeomorphism: bijection, continuous in both directions Theorem Metric X = fx1;x2;:::g,Y = fy1;y2;:::g no isolated points =) X ≈ Y Proof: Elementary proof with finite recursion I Toolbox I ball: B(x;r) = fy : d(x;y) < rg; r > 0 I open set: union of balls I isolated: fxg is a ball I continuous function: e;d I D = fdX (xm;xn): m 6= ng [ fdY (ym;yn): m 6= ng I D ⊂ (0;¥) countable I S ⊂ X;Y : next(S) = xk where k = minfi : xi 2 Sg Theorem Metric X = fx1;x2;:::g,Y = fy1;y2;:::g no isolated points =) X ≈ Y Proof: Elementary proof with finite recursion I Toolbox I ball: B(x;r) = fy : d(x;y) < rg; r > 0 I open set: union of balls I isolated: fxg is a ball I continuous function: e;d I homeomorphism: bijection, continuous in both directions I D = fdX (xm;xn): m 6= ng [ fdY (ym;yn): m 6= ng I D ⊂ (0;¥) countable I S ⊂ X;Y : next(S) = xk where k = minfi : xi 2 Sg Theorem Metric X = fx1;x2;:::g,Y = fy1;y2;:::g no isolated points =) X