Subspaces, Products, and Quotients

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✓ 19 results · all verified · 11 also independently AI-judged

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1 · Prerequisites

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Theorem

For A ⊆ S ⊆ X the closure of A in S is Ā^X ∩ S, while the interior only contains int^X(A) ∩ S, with equality when S is open; and a dense subset of X traces to a dense subset of every open S

Definition

Hereditary, open-hereditary and closed-hereditary properties of topological spaces

Corollary

Every subspace of a metrizable space is metrizable and every subspace of a first countable space is first countable, the metric case being the subspace metric already identified with the subspace topology

Definition

The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology

Theorem

Characteristic properties: a map into a space with the initial topology is continuous iff every composite with the defining family is, a map out of a space with the final topology is continuous iff every composite with the defining family is, and the two topologies are respectively the coarsest and the finest making that family continuous

Definition

The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space

Theorem

A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice

Theorem

The box topology is finer than the product topology, the two agree for a finite index set in ZF, and, assuming the Axiom of Choice for nonempty factors, the box topology is strictly finer whenever infinitely many factors have a nonempty proper open subset

Theorem

Products commute with subspaces; for infinite nonempty families, the closure identity ∏ Aᵢ=∏ Aᵢ uses the Axiom of Choice

Lemma

For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space

Definition

The disjoint union (coproduct) ⨆ᵢ Xᵢ with the final topology of the canonical injections: a set is open exactly when each of its traces is

Theorem

A map out of a disjoint union is continuous iff each of its restrictions is; the canonical injections are open and closed embeddings; and each summand is clopen in the union

Definition

The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

Theorem

For a quotient map q : X → Y, a map out of Y is continuous iff its composite with q is; a continuous map on X constant on the fibres of q factors uniquely through q; and a composite of quotient maps is a quotient map

Lemma

A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps

Theorem

Every quotient map q : X → Y induces a homeomorphism from X modulo the relation "q agrees" onto Y, so up to homeomorphism the quotient maps out of X are exactly the canonical projections

Definition

The adjunction space Y ∪_f X glued along a continuous map, and, for a nonempty space, the cone and the suspension as quotients of X × [0,1]

Definition

Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not

Remark

What the theory of these constructions still owes at this point in the reading order: preservation of quotient maps under products, separation beyond Hausdorff, and the invariants that tell the glued spaces apart

Example

ℝ/ℤ: the quotient map is open, and the quotient is homeomorphic to [0,1] with its endpoints identified

Example

The square with opposite edges identified is homeomorphic to the product (ℝ/ℤ) × (ℝ/ℤ)

Example

The cylinder and the Mobius band as quotients of the square by (0,y) ∼ (1,y) and by (0,y) ∼ (1, 1-y), both by a closed quotient map

False statement

Subspaces, Products, and Quotients

3 · Items

For A ⊆ S ⊆ X the closure of A in S is Ā^X ∩ S, while the interior only contains int^X(A) ∩ S, with equality when S is open; and a dense subset of X traces to a dense subset of every open S

Hereditary, open-hereditary and closed-hereditary properties of topological spaces

Every subspace of a metrizable space is metrizable and every subspace of a first countable space is first countable, the metric case being the subspace metric already identified with the subspace topology

The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology

The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space

A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice

The box topology is finer than the product topology, the two agree for a finite index set in ZF, and, assuming the Axiom of Choice for nonempty factors, the box topology is strictly finer whenever infinitely many factors have a nonempty proper open subset

Products commute with subspaces; for infinite nonempty families, the closure identity ∏ Aᵢ=∏ Aᵢ uses the Axiom of Choice

For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space

The disjoint union (coproduct) ⨆ᵢ Xᵢ with the final topology of the canonical injections: a set is open exactly when each of its traces is

A map out of a disjoint union is continuous iff each of its restrictions is; the canonical injections are open and closed embeddings; and each summand is clopen in the union

The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

For a quotient map q : X → Y, a map out of Y is continuous iff its composite with q is; a continuous map on X constant on the fibres of q factors uniquely through q; and a composite of quotient maps is a quotient map

A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps

Every quotient map q : X → Y induces a homeomorphism from X modulo the relation "q agrees" onto Y, so up to homeomorphism the quotient maps out of X are exactly the canonical projections

The adjunction space Y ∪_f X glued along a continuous map, and, for a nonempty space, the cone and the suspension as quotients of X × [0,1]

Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not

What the theory of these constructions still owes at this point in the reading order: preservation of quotient maps under products, separation beyond Hausdorff, and the invariants that tell the glued spaces apart

ℝ/ℤ: the quotient map is open, and the quotient is homeomorphic to [0,1] with its endpoints identified

The square with opposite edges identified is homeomorphic to the product (ℝ/ℤ) × (ℝ/ℤ)

The cylinder and the Mobius band as quotients of the square by (0,y) ∼ (1,y) and by (0,y) ∼ (1, 1-y), both by a closed quotient map

FALSE: the product topology and the box topology agree on every product

FALSE: ∏ᵢ Uᵢ is open in the product topology whenever every Uᵢ is open

FALSE: the projections of a product are closed maps

FALSE: every quotient map is an open map

FALSE: a quotient of a Hausdorff space is Hausdorff