TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Statement

Let $X$ be a set and let $I$ be an index set.

Initial. Let $(Y_{i}, T_{i})$ be spaces and $f_{i} : X \to Y_{i}$ functions, and give $X$ the initial topology $T^{i n}$ of the family $(f_{i})_{i \in I}$ (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). Then:

Every $f_{i}$ is continuous for $T^{i n}$ , and $T^{i n}$ is the coarsest topology on $X$ with that property: every topology on $X$ making all the $f_{i}$ continuous contains $T^{i n}$ .
Characteristic property. For every space $Z$ and every function $h : Z \to X$ , $h is continuous ⟺ f_{i} \circ h is continuous for every i \in I .$

Final. Let $(Z_{i}, S_{i})$ be spaces and $g_{i} : Z_{i} \to X$ functions, and give $X$ the final topology $T^{f i n}$ of the family $(g_{i})_{i \in I}$ . Then:

Every $g_{i}$ is continuous for $T^{f i n}$ , and $T^{f i n}$ is the finest topology on $X$ with that property: every topology on $X$ making all the $g_{i}$ continuous is contained in $T^{f i n}$ .
Characteristic property. For every space $W$ and every function $k : X \to W$ , $k is continuous ⟺ k \circ g_{i} is continuous for every i \in I .$

Claims 2 and 4 determine their topologies: a topology on $X$ satisfying claim 2 for every $Z$ and $h$ must equal $T^{i n}$ , and likewise for claim 4, by the argument recorded in the remarks.

Facts & Assumptions

Given: A set $X$ ; spaces $(Y_{i}, T_{i})$ with functions $f_{i} : X \to Y_{i}$ ; spaces $(Z_{i}, S_{i})$ with functions $g_{i} : Z_{i} \to X$ ; a space $Z$ with a function $h : Z \to X$ and a space $W$ with a function $k : X \to W$ . Preimages satisfy $(u \circ v)^{- 1} [T] = v^{- 1} [u^{- 1} [T]]$ for composable functions $u, v$ and every subset $T$ of the target.

[A1]

$T^{i n} = ⟨ G ⟩$ where $G : = {f_{i}^{- 1} [V] : i \in I, V \in T_{i}}$ , and $T^{f i n} = {U \subseteq X : g_{i}^{- 1} [U] \in S_{i} for every i}$ (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).

[L1]

$⟨ G ⟩$ is a topology containing $G$ and contained in every topology containing $G$ ; and $G$ is a subbasis for it (Basis and subbasis for a topology, and the topology generated by a family of sets, A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis).

[L2]

A map of spaces is continuous if and only if preimages of open sets are open, and if and only if preimages of the members of some subbasis of the target are open (For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ , clauses (b) and (d); Continuity of a map of topological spaces at a point and globally).

[L3]

A composite of continuous maps is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous, claim 1).

[L4]

A topology contains $\emptyset$ and the whole set and is closed under arbitrary unions and binary intersections (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

Proof

technique · direct

1.1

Each $f_{i}$ is continuous for $T^{i n}$ : for $V \in T_{i}$ the set $f_{i}^{- 1} [V]$ lies in $G \subseteq ⟨ G ⟩ = T^{i n}$ , so preimages of open sets are open.

A1L1L2

1.2

Let $T^{'}$ be a topology on $X$ making every $f_{i}$ continuous. Then $G \subseteq T^{'}$ by [L2], so $T^{i n} = ⟨ G ⟩ \subseteq T^{'}$ by [L1].

A1L1L2

1.3

Each $g_{i}$ is continuous for $T^{f i n}$ : if $U \in T^{f i n}$ then $g_{i}^{- 1} [U] \in S_{i}$ by the defining condition.

A1L2

1.4

Let $T^{''}$ be a topology on $X$ making every $g_{i}$ continuous, and let $U \in T^{''}$ . Then $g_{i}^{- 1} [U] \in S_{i}$ for every $i$ by [L2], so $U \in T^{f i n}$ ; hence $T^{''} \subseteq T^{f i n}$ .

A1L2

1.5

Assume every $f_{i} \circ h$ is continuous. For $i \in I$ and $V \in T_{i}$ one has $h^{- 1} [f_{i}^{- 1} [V]] = (f_{i} \circ h)^{- 1} [V]$ , which is open in $Z$ ; so preimages under $h$ of all members of $G$ are open, and $G$ is a subbasis for $T^{i n}$ , so $h$ is continuous by clause (d) of [L2].

givenA1L1L2

1.6

Assume every $k \circ g_{i}$ is continuous. For $V$ open in $W$ and each $i$ one has $g_{i}^{- 1} [k^{- 1} [V]] = (k \circ g_{i})^{- 1} [V]$ , which is open in $Z_{i}$ ; so $k^{- 1} [V] \in T^{f i n}$ by the defining condition, and $k$ is continuous by clause (b) of [L2].

givenA1L2

2.1

If $h : Z \to X$ is continuous then each $f_{i} \circ h$ is continuous, and if $k : X \to W$ is continuous then each $k \circ g_{i}$ is continuous, in both cases as a composite of continuous maps, the $f_{i}$ being continuous by step 1.1 and the $g_{i}$ by step 1.3.

step 1.1step 1.3L3

2.2

Steps 1.1 and 1.2 are claim 1, and steps 1.3 and 1.4 are claim 3.

step 1.1step 1.2step 1.3step 1.4L4

3.1

Step 2.1 gives the forward implications of claims 2 and 4, and steps 1.5 and 1.6 give the reverse implications; so claims 2 and 4 hold, and with step 2.2 all four claims are proved.

step 2.1step 1.5step 1.6step 2.2∎

Remarks

The characteristic property pins the topology down. Suppose two topologies $T_{1}$ and $T_{2}$ on $X$ both satisfy claim 2 for every space $Z$ and every function $h$ . Apply claim 2 for $T_{1}$ to $Z = (X, T_{2})$ and $h = i d$ : the composites $f_{i}$ are continuous on $(X, T_{2})$ by claim 2 for $T_{2}$ applied to the identity of $(X, T_{2})$ , so the identity $(X, T_{2}) \to (X, T_{1})$ is continuous, that is $T_{1} \subseteq T_{2}$ . Exchanging the roles gives equality. The same argument with the arrows reversed does claim 4.
Only continuity of the composites is tested, never their openness. Claim 2 says nothing about whether $h$ is open or closed, and claim 4 says nothing about $k$ ; the constructions below acquire such properties one at a time and each is proved where it is used.
The one-element family is not a degenerate case but the main one. The subspace topology is the initial topology of a single inclusion and the quotient topology is the final topology of a single surjection (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), so claims 2 and 4 with $I$ a one-element set already carry the characteristic properties of both.

Depends on

Used by

Dependency tree · two levels

14 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

Initial topology (Wikipedia) (standard reference, not scraped)
Final topology (Wikipedia) (standard reference, not scraped)
J. Munkres, Topology, 2nd ed., §19 (standard reference, not scraped)