An Illustrated Introduction to Topology and Homotopy by Sasho Kalajdzievski

By Sasho Kalajdzievski

An Illustrated creation to Topology and Homotopy explores the great thing about topology and homotopy thought in a right away and fascinating demeanour whereas illustrating the facility of the speculation via many, frequently impressive, functions. This self-contained booklet takes a visible and rigorous strategy that includes either large illustrations and entire proofs.

The first a part of the textual content covers easy topology, starting from metric areas and the axioms of topology via subspaces, product areas, connectedness, compactness, and separation axioms to Urysohn’s lemma, Tietze’s theorems, and Stone-Čech compactification. targeting homotopy, the second one half starts off with the notions of ambient isotopy, homotopy, and the elemental workforce. The publication then covers uncomplicated combinatorial staff idea, the Seifert-van Kampen theorem, knots, and low-dimensional manifolds. The final 3 chapters talk about the idea of masking areas, the Borsuk-Ulam theorem, and functions in workforce concept, together with quite a few subgroup theorems.

Requiring just some familiarity with workforce concept, the textual content encompasses a huge variety of figures in addition to numerous examples that exhibit how the idea may be utilized. each one part starts off with short ancient notes that hint the expansion of the topic and ends with a suite of workouts.

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Example text

We will invoke the second, stronger version several times later on. Zorn’s Lemma 1. If every chain in an ordered nonempty set A has an upper bound in A, then A contains a maximal element. Zorn’s Lemma 2. If every chain in an ordered nonempty set A has an upper bound in A, then for every a ∈ A there is a maximal element m ∈ A such that a ≤ m. Proof. ) Take any a ∈ A and consider the set Y = { y ∈ A : a ≤ y} . Clearly a ∈Y , so Y ≠ ∅ . By assumption, every chain of elements in Y (being at the same time a chain of elements in A) has an upper bound in A.

Maximal elements need not be unique. For example, A = {a, b, c} with an order < defined by a < b, a < c , has two maximal elements, b and c. A chain in A is any subset B of A that inherits a linear order from the order of A. For example, with the partial order on A = {a, b, c} as above, the set {a, b} is a chain in A. An element u ∈ A is an upper bound of subset B of A if b ≤ u for every b ∈B . Considering again the set A = {a, b, c} with the given order, we notice that every element of A is an upper bound for {a}, and that {b, c} has no upper bound.

Generalize: Let Xi be spaces, let Y be a set and let fi : Xi → Y , i ∈I , be mappings. Show that T = U ⊂ Y : for every i ∈I , fi −1 (U ) ⊂ Xi is a topology over Y. Show that the (Euclidean) metric space topology and the (usual) ordered topology over » are the same. Show that the order topology defined in Example 8 is indeed a topology. Let τ be the collection defined in Example 8, except that we do not require that X be in τ. Show that in that case τ is not necessarily a topology. 2 Some Basic Notions A brief historical note: Bernard Bolzano (1781–1848) proved Theorem 5 in the 1830s.

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