False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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FALSE: every quotient map is an open map

Statement

False claim: every quotient map $q : X \to Y$ (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection) is an open map, that is, carries open subsets of $X$ to open subsets of $Y$ (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

The converse implication is the one that holds: a continuous open surjection is a quotient map (A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps, clause 1). The claim above fails for the cheapest identification there is, collapsing a closed interval of $R$ to a point. Take $X : = R$ with its usual topology (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not), $B : = [0, 1]$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), and let

$q : R \to R / B$

be the canonical projection of the quotient that identifies all of $B$ to one point and identifies nothing else (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection). Then $q$ is a quotient map by construction, and $q [(- 1, 1 / 2)]$ is not open.

Facts & Assumptions

Given: $R$ with its usual topology; $B = [0, 1]$ ; the equivalence relation on $R$ whose classes are $B$ and the singletons ${t}$ for $t \notin B$ ; the quotient $R / B$ with the quotient topology and its canonical projection $q$ ; and the set $U : = (- 1, 1 / 2)$ .

[A1]

$q$ is a surjection, the topology of $R / B$ is the quotient topology of $q$ , and consequently $V \subseteq R / B$ is open exactly when $q^{- 1} [V]$ is open in $R$ ; so $q$ is a quotient map (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[A2]

$A \subseteq R$ is saturated for $q$ exactly when $A \cap B$ is $\emptyset$ or $B$ , and $q^{- 1} [q [A]]$ is the saturation of $A$ (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[A3]

$f$ is an open map when images of open sets are open (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

[L1]

$(a, b) = {t : a < t < b}$ and $[0, 1] = {t : 0 \leq t \leq 1}$ ; $U \subseteq R$ is open in the usual topology exactly when every point of $U$ has a bounded open interval around it inside $U$ , and every bounded open interval is open (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not).

[L2]

A continuous open surjection is a quotient map (A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps, clause 1).

[L3]

A topology is a family of subsets of the underlying set (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

Refutation

technique · direct

1.1

$U = (- 1, 1 / 2)$ is open in $R$ , being a bounded open interval.

1.2

$U \cap B = [0, 1 / 2)$ , which is neither $\emptyset$ , since it contains $0$ , nor $B$ , since $1 \in B$ and $1 \notin [0, 1 / 2)$ ; so $U$ is not saturated.

A2L1

1.3

$(- 1, 1]$ is not open in $R$ : for every $r > 0$ the interval $(1 - r, 1 + r)$ contains $1 + r / 2$ , which satisfies $1 + r / 2 > 1$ and so lies outside $(- 1, 1]$ ; hence no bounded open interval around $1$ lies inside $(- 1, 1]$ .

2.1

$q^{- 1} [q [U]] = U \cup B = (- 1, 1]$ : the saturation of $U$ adds to $U$ exactly the class of each of its points, and the only non-singleton class meeting $U$ is $B$ itself, by step 1.2.

step 1.2A2L1

3.1

By step 2.1 and step 1.3 the set $q^{- 1} [q [U]]$ is not open in $R$ , so $q [U]$ is not open in $R / B$ by [A1].

step 2.1step 1.3A1L3

4.1

By [A1] the map $q$ is a quotient map, and by step 1.1 and step 3.1 it carries the open set $U$ to a set that is not open; so $q$ is not an open map by [A3], and the claim is false.

step 1.1step 3.1A1A3L2∎

Remarks

The obstruction is saturation, and it is the general one. A quotient map is open exactly when the saturation of every open set is open (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection), and collapsing a set with nonempty interior destroys that: an open set that meets $B$ without containing it acquires the whole of $B$ in its saturation, and $B$ has boundary points.
Closedness fails independently. The map above happens to be closed, since the saturation of a closed set $F$ is $F$ or $F \cup B$ , both closed; so this witness separates "quotient map" from "open map" only. A quotient map that is neither open nor closed needs a different construction, and one is worked on the companion page as On $A = ([0, \infty) \times R) \cup (R \times {0})$ the first projection is a quotient map, by the section $x \mapsto (x, 0)$ , and is neither open nor closed ↗.
Why the converse direction is nevertheless useful. Most quotients that are identified with a known space in practice are open quotient maps, because their equivalence relation comes from translating by the elements of a group, and translation is a homeomorphism; both the circle and the torus on the companion page are of that kind.

Depends on

Used by

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

Dependency tree · two levels

42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Quotient space (topology) (Wikipedia) (standard reference, not scraped)
Open and closed maps (Wikipedia) (standard reference, not scraped)