Subspaces, Products, and Quotients: Examples and Counterexamples

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1 · Prerequisites

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Example

ℝⁿ as the product of n copies of the real line: the product topology is the Euclidean topology and the projections are continuous, open and surjective

Counterexample

The hyperbola {(x,y) : xy = 1} is closed in ℝ² and its image under the first projection is ℝ ∖ {0}, which is not closed

Counterexample

The diagonal x ↦ (x,x,…) from ℝ into ℝ^ℕ is continuous for the product topology and not for the box topology

Example

The Cantor set is homeomorphic to {0,1}^ℕ with the product of discrete topologies, the ternary digits being the coordinates

Example

The Hilbert cube [0,1]^ℕ with the product topology is metrizable, by d(x,y) = ∑ₖ |xₖ - yₖ| / 2^ k+1

Counterexample

On A = ([0,∞) × ℝ) ∪ (ℝ × {0}) the first projection is a quotient map, by the section x ↦ (x,0), and is neither open nor closed

Counterexample

Two copies of ℝ glued along ℝ ∖ {0} give a non-Hausdorff quotient of a metrizable space, by an open quotient map

Counterexample

ℝ/ℚ carries the indiscrete topology, although ℝ is metrizable and the quotient has more than one point

Example

The Sorgenfrey plane: the product of two half-open-interval lines has the rectangles [a,b) × [c,d) as a basis and ℚ × ℚ as a countable dense subset

Counterexample

Subspaces, Products, and Quotients: Examples and Counterexamples

3 · Items

ℝⁿ as the product of n copies of the real line: the product topology is the Euclidean topology and the projections are continuous, open and surjective

The hyperbola {(x,y) : xy = 1} is closed in ℝ² and its image under the first projection is ℝ ∖ {0}, which is not closed

The diagonal x ↦ (x,x,…) from ℝ into ℝ^ℕ is continuous for the product topology and not for the box topology

The Cantor set is homeomorphic to {0,1}^ℕ with the product of discrete topologies, the ternary digits being the coordinates

The Hilbert cube [0,1]^ℕ with the product topology is metrizable, by d(x,y) = ∑ₖ |xₖ - yₖ| / 2^ k+1

On A = ([0,∞) × ℝ) ∪ (ℝ × {0}) the first projection is a quotient map, by the section x ↦ (x,0), and is neither open nor closed

Two copies of ℝ glued along ℝ ∖ {0} give a non-Hausdorff quotient of a metrizable space, by an open quotient map

ℝ/ℚ carries the indiscrete topology, although ℝ is metrizable and the quotient has more than one point

The Sorgenfrey plane: the product of two half-open-interval lines has the rectangles [a,b) × [c,d) as a basis and ℚ × ℚ as a countable dense subset

The antidiagonal {(x,-x)} is an uncountable discrete subspace of the Sorgenfrey plane, so having a countable dense subset is not a hereditary property