Path-connected space

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

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 — Require continuity of the full path.
• Connected topological space — Retain the forward implication and converse boundary.
• Real number — Use the unit interval.
• Usual order on the real numbers — Specify unit parameters and range bounds.
• Finite product topology — Use the box path codomain.
• Intermediate value theorem — Use the gap/connected-image obstruction.
• Function composition — Retain concatenation and continuous images.
• 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.