Quotient topology

Mathematical statement

In finite models first qualify q:X→Y as a total surjection and enumerate every fiber. Define tau_q={U⊆Y:q^(-1)(U) is open in X}; enumerate the entire family and verify its topology axioms, continuity of q and the final property: h:Y→Z is continuous if and only if h∘q is continuous. Only saturated open subsets of X can be full preimages of quotient opens; images of arbitrary opens need not be open. For every total map, including a nonsurjection, the defining preimage-if-and-only-if condition still uniquely determines the final topology on Y. Requiring only continuity of q is generally weaker and does not uniquely determine that topology. Unhit points do not invalidate this definition, but under the convention used in this node a nonsurjection is not called a quotient map.

Representative formula (exact TeX)

Q(X,q,Y)=t

Assumptions

• X and Y are finite unique carriers, with q explicitly a total surjection in ordinary quotient families.
• Preimages, fibers, saturations and topology families are completely enumerated.

Symbol scope

• Q — quotient-topology operator
• X — source topological space
• q — surjective quotient map
• Y — quotient carrier
• t — quotient topology

Intuition, definition, and misconceptions

Under the quotient-map convention here, first qualify a total surjection and its fibers; for a general total map, the final topology is still defined by the full preimage-if-and-only-if condition.

Define opens by complete preimages, not arbitrary open images.

A finite final-property audit does not replace the general proof.

Using open images of q as the quotient topology, or confusing continuity alone with the full preimage-if-and-only-if definition. That definition uniquely determines a final topology even for a nonsurjection, although this node reserves the name quotient map for surjections.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Topological space — Qualify source and target topology axioms.
• Quotient set — Use fibers and equivalence classes.
• Surjectivity and codomain coverage — Retain the quotient-map carrier obligation.
• Continuous maps between topological spaces — Audit q and the final property.
• Function composition — Compare h with h∘q.

Verification boundary

Finite tables certify the specified quotient and do not automatically preserve Hausdorff separation. The Chinese prose about an unhit point not being determined needs final semantic alignment: a non-surjective map still has a uniquely specified final topology under the full preimage iff condition, as the exact boundary records. No original source text or formula is changed.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add identification spaces and quotient separation conditions.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.