# Path-connected space

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A space is path connected when every pair of its points is joined by a continuous map from the unit interval.

mko-path-connected-space · definition · v0.10.0



## Mathematical statement

A path is a continuous gamma:[0,1]→X with gamma(0)=x and gamma(1)=y. Exact families check linear paths in real intervals and finite-dimensional boxes, together with piecewise concatenation, including endpoints, full range containment and the shared join. Connected-image or IVT reasoning excludes a path across a disconnected gap or between the two points of a discrete space. Finite samples, continuity-free endpoint data or a path for one pair do not establish the universal property. Path connectedness implies connectedness, while the general converse fails.





## Representative formula (exact TeX)

H(X,x,y)=g





## Assumptions


- The path domain is the closed unit interval with its usual topology.
- Every candidate path needs continuity, the correct endpoints and whole-range containment.





## Symbol scope


- H — path-connectedness construction
- X — topological space
- x — path start
- y — path end
- g — continuous path





## Intuition, definition, and misconceptions

Endpoints, continuity and whole-range containment are all necessary.

Convex intervals and boxes give explicit affine paths; concatenation checks its join.

One pair or finite samples do not prove universal path connectedness.

A point list, graph walk or one joinable pair is not the topological path-connectedness property.





## Adopted derivation guide

[Universal paths and concatenation in a convex box](https://ocme.evemisslab.com/en/derivations/derivation-mko-path-connected-space-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/mko/mko-topological-continuity/) — Require continuity of the full path.
- [Connected topological space](https://ocme.evemisslab.com/en/mko/mko-connected-topological-space/) — Retain the forward implication and converse boundary.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — Use the unit interval.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Specify unit parameters and range bounds.
- [Finite product topology](https://ocme.evemisslab.com/en/mko/mko-finite-product-topology/) — Use the box path codomain.
- [Intermediate value theorem](https://ocme.evemisslab.com/en/mko/mko-intermediate-value-theorem/) — Use the gap/connected-image obstruction.
- [Function composition](https://ocme.evemisslab.com/en/mko/mko-composition/) — Retain concatenation and continuous images.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Use affine and piecewise parameterizations.





## Verification boundary

Exact affine witnesses certify the specified convex spaces only. The named sine-curve boundary records the connected-not-path scope without constructing or proving the complete analytic example in this finite family.

not_formalized

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





## Next obligation

Add local path connectedness and an entry to the fundamental group.





## 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/path-connected-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-path-connected-space/index.md)