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 = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

no isolated points =⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

=⇒ X ≈ Y Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

Proof:

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S}

Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable Elementary proof with finite recursion

I Toolbox I ball: B(x,r) = {y : d(x,y) < r}, r > 0 I open set: union of balls I isolated: {x} is a ball I continuous function: ε,δ I homeomorphism: bijection, continuous in both directions Theorem Metric X = {x1,x2,...},Y = {y1,y2,...} no isolated points =⇒ X ≈ Y Proof:

I D = {dX (xm,xn): m 6= n} ∪ {dY (ym,yn): m 6= n} I D ⊂ (0,∞) countable

I S ⊂ X,Y : next(S) = xk where k = min{i : xi ∈ S} The Splitter

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ⊂ X × Q ≈ Q

QED! Proof. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ⊂ X × Q ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ⊂ X × Q ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ⊂ X × Q ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ⊂ X × Q ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. X × Q: Countable, no isolated points X ≈ X × {0} ⊂ X × Q ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| X ≈ X × {0} ⊂ X × Q ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points ⊂ X × Q ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ≈ Q

QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ⊂ X × Q QED!

Corollary Every countable metric space is homeomorphic to a subspace of Q. Proof. X countable. X × Q: metric ρ((x,p),(y,q)) = d(x,y) + |p − q| Countable, no isolated points X ≈ X × {0} ⊂ X × Q ≈ Q References

Dasgupta, A. Countable metric spaces without isolated points. Topology Explained. June 2005, published by Topology Atlas. Accessed 7 June 2021 at http://dasgupab.faculty.udmercy.edu. Dashiell, F., Countable metric spaces without isolated points. Amer. Math. Monthly 128(3) (2021), 265–267.