Experiments in Topology by Stephen Barr

By Stephen Barr

With this booklet and a sq. sheet of paper, the reader could make paper Klein bottles; then by way of intersecting or slicing the bottle, make Moebius strips. Conical Moebius strips, projective planes, the main of map coloring, the vintage challenge of the Koenigsberg bridges and different elements of topology are clearly explained.

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Consequently, σ ∧. γ ∈ id∧ ( ). v. We have σ ∧. (τ ∧. γ ) ≤ σ ∧ (τ ∧. γ ) ≤ σ ∧ γ ) = (σ ∧ τ ) ∧ γ . In particular σ ∧. (τ ∧. γ ) ≤ σ ∧ τ and σ ∧. (τ ∧. γ ) ≤ γ , thus, because σ ∧. (τ ∧. γ ) is idempotent (see iv) we obtain σ ∧. γ ) ≤ σ ∧. τ and σ ∧. (τ ∧. γ ) ≤ γ , hence, again by iv σ ∧. (τ ∧. γ ) ≤ (σ ∧. )∧. γ . The converse inequality may be derived in formally the same way. vi. If x ∈ id∧ ( ) and x ≤ z, then x = x∧. z follows. A fortiori x ∨ (x∧. z)=x=x∧. z, and moreover x ∨ (z∧. x) = z∧.

X ∨ z) ≥ x ∨ (y∧ . z). The lattice id∧ ( ),∧ . , ∨ will be denoted by S L( ), and it is called the commutative shadow (lattice) of . The term commutative is to the point because it is clear from the definition that σ ∧ . τ = τ∧ . σ. 1 S L( ) satisfies the modular inequality. Proof Take σ, τ, γ in S L( ) and assume that σ ≤ γ . We have to establish that σ ∨ (τ ∧. γ ) ≤ (σ ∨ τ )∧. γ . The fact that σ ∨ (τ ∧. γ ) is idempotent, combined with σ ∨ (τ ∧. γ ) ≤ σ ∨ τ and σ ∨(τ ∧. γ ) ≤ γ , entails σ ∨(τ ∧.

N ≤ λn such that (∨ λα ) ∧ µ ⊂ ∨(λα ∧ µ). 6 Suppose that is converging-distributive. i. Any minimal-point is a point. ii. Any [ p] ∈ C( ) is a point if and only if it is a ∨-irreducible element in C( ). Proof i. If [A] represents a minimal-point, then A is a maximal filter (= ). Assume that ∩{Aα , α ∈ A} ⊂ A and Aα ⊂ A for all α. Pick aα ∈ Aα − A for every α ∈ A. Look at Bα = {aα ∧ a, a ∈ A}; this is obviously a directed set in . Moreover B α ⊃ A because for every a ∈ A there is aα ∧ a ∈ Bα such that aα ∧ a ≤ a.

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