A General Topology Workbook by Iain Adamson

By Iain Adamson

This e-book has been known as a Workbook to make it transparent from the beginning that it's not a traditional textbook. traditional textbooks continue by means of giving in each one part or bankruptcy first the definitions of the phrases for use, the recommendations they're to paintings with, then a few theorems regarding those phrases (complete with proofs) and eventually a few examples and routines to check the readers' realizing of the definitions and the theorems. Readers of this ebook will certainly locate the entire traditional constituents--definitions, theorems, proofs, examples and exercises­ yet now not within the traditional association. within the first a part of the publication might be discovered a short evaluate of the elemental definitions of common topology interspersed with a wide num­ ber of routines, a few of that are additionally defined as theorems. (The use of the notice Theorem isn't really meant as a sign of trouble yet of significance and value. ) The routines are intentionally no longer "graded"-after the entire difficulties we meet in mathematical "real life" don't are available order of trouble; a few of them are extremely simple illustrative examples; others are within the nature of educational difficulties for a conven­ tional direction, whereas others are rather tough effects. No ideas of the routines, no proofs of the theorems are incorporated within the first a part of the book-this is a Workbook and readers are invited to attempt their hand at fixing the issues and proving the theorems for themselves.

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1 Theorem. (i) Every functor preserves commutative diagrams. (ii) Every faithful functor reflects commutative diagrams. Proof. Let F : C → D be a functor. Let α1 , . . , αn and β1 , . . , βm be morphisms in C such that α, β : x → y for some objects x and y of C, where α = αi and β = βi . If α = β, then F (αi ) = F ( αi ) = F (α) = F (β) = F ( βi ) = F (βi ). This proves (i). Now assuming that F is faithful and F (αi ) = F (βi ), we get (rearranging the string of equalities above) F (α) = F (β), so that α = β.

5 and the category D = FVectK of finite-dimensional vector spaces over K are equivalent. We get a functor F : C → D by defining F (n) = K n (= space of ndimensional column vectors over K) for each object n of C and F (A) = µA for each morphism A : n → n in C, where µA : K n → K n is the linear map given by µA (v) = Av. Choose a basis for each finite-dimensional vector space, with the choice for each vector space K n (n ∈ N) being the standard basis. We get a functor G : D → C by defining G(V ) = dim V for each object V of D and G(α) = Mα for each morphism α : V → V in D, where Mα is the matrix of α relative to the chosen bases of V and V .

Conversely, the restriction of any functor F : CM → CN to morphisms yields a homomorphism M → N . 4 Example (Power set functor) The power set functor P : Set → Set is defined as follows: For an object X, put P (X) = { S| S ⊆ X} (the power set of X), and for a morphism α : X → Y define P (α) : P (X) → P (Y ) by P (α)(S) = α[S]. (Cf. 5 Example (Fundamental group functor) Let X be a topological space and let p be a fixed point of X. A “loop at p” is a continuous map 43 γ : [0, 1] → X such that γ(0) = p and γ(1) = p.

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