Connectedness

When to Use

Use this skill when working on connectedness problems in topology.

Decision Tree

1. Is X connected?

- Strategy 1 - Contradiction: Assume X = U union V where U, V are disjoint, non-empty, and open Derive a contradiction - Strategy 2 - Path connectedness: Show for all x,y in X, exists continuous path f: [0,1] -> X with f(0)=x, f(1)=y - Strategy 3 - Fan lemma: If {Ai} are connected sharing a common point, then union Ai is connected

1. Connectedness Proofs

- Show no separation exists - z3solve.py prove "noseparation" - Use intermediate value theorem for R subsets

1. Path Connectedness

- Construct explicit path: f(t) = (1-t)x + ty for convex sets - sympy_compute.py simplify "(1-t)x + ty" to verify path

1. Components

- Connected component: maximal connected subset containing x - Path component: maximal path-connected subset containing x

Tool Commands

Z3NoSeparation

uv run python -m runtime.harness scripts/z3_solve.py prove "no_separation"

Sympy_Path

uv run python -m runtime.harness scripts/sympy_compute.py simplify "(1-t)*x + t*y"

Z3_Ivt

uv run python -m runtime.harness scripts/z3_solve.py prove "intermediate_value"

Key Techniques

From indexed textbooks:

• [Introduction to Topological Manifolds... (Z-Library)] Connectedness One of the most important elementary facts about continuous functions is the intermediate value theorem: If f is a continuous real-valued function dened on a closed bounded interval [a, b], then f takes on every value be- tween f (a) and f (b). The key idea here is the “connectedness” of intervals. In this section we generalize this concept to topological spaces.
• [Topology (Munkres, James Raymond) (Z-Library)] A b× lb× cb×0π1(A)×0π1(A)×0 156ConnectednessandCompactnessCh. DenetheunitballBninRnbytheequationBn={x|x≤1},wherex=(x1,. Theunitballispathconnected;givenanytwopointsxandyofBn,thestraight-linepathf:[0,1]→Rndenedbyf(t)=(1−t)x+tyliesinBn.
• [Introduction to Topological Manifolds... (Z-Library)] Thanks are due also to Mary Sheetz, who did an excellent job producing some of the illustrations under the pressures of time and a nicky author. My debt to the authors of several other textbooks will be obvious to anyone who knows those books: William Massey’s Algebraic Topology: An Introduction [Mas89], Allan Sieradski’s An Introduction to Topology and Homotopy [Sie92], Glen Bredon’s Topology and Geometry, and James Munkres’s Topology: A First Course [Mun75] and Elements of Algebraic Topology [Mun84] are foremost among them. Finally, I would like to thank my wife, Pm, for her forbearance and unagging support while I was spending far too much time with this book Preface and far too little with the family; without her help I unquestionably could not have done it.
• [Topology (Munkres, James Raymond) (Z-Library)] TheunionofacollectionofconnectedsubspacesofXthathaveapointincommonisconnected. Let{Aα}beacollectionofconnectedsubspacesofaspaceX;letpbeapointofAα. WeprovethatthespaceY=Aαisconnected.
• [Introduction to Topological Manifolds... (Z-Library)] Conversely, if X is disconnected, we can write X = U ∪ V where U and V are nonempty, open, and disjoint. This implies that U is open, closed, not empty, and not equal to X. Main Theorem on Connectedness).

Cognitive Tools Reference

See .maestro/skills/math-mode/SKILL.md for full tool documentation.