By Robert A. Conover

**Publish yr note:** initially released in 1975

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Students needs to turn out all the theorems during this undergraduate-level textual content, which positive factors large outlines to aid in research and comprehension. Thorough and well-written, the therapy offers adequate fabric for a one-year undergraduate direction. The logical presentation anticipates students' questions, and entire definitions and expositions of issues relate new ideas to formerly mentioned subjects.

Most of the fabric makes a speciality of point-set topology except for the final bankruptcy. subject matters comprise units and capabilities, limitless units and transfinite numbers, topological areas and easy thoughts, product areas, connectivity, and compactness. extra matters contain separation axioms, whole areas, and homotopy and the basic team. quite a few tricks and figures light up the text.

**Read or Download A First Course in Topology: An Introduction to Mathematical Thinking PDF**

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**Extra info for A First Course in Topology: An Introduction to Mathematical Thinking**

**Example text**

If (C, F ) is a concrete category, then (C op , F ) is a concrete category for some (faithful) functor F : C op → Set. Proof. Let (C, F ) be a concrete category. The functor F : C → Set induces a faithful (covariant) functor F : C op → Setop (same object map and same morphism map). 3 can be regarded as a covariant functor P : Setop → Set. Claim: P is faithful. Let α, β : Y → X be morphisms in Setop and assume that α = β. Then the functions α, β : X → Y are not equal, implying that α(x) = β(x) for some x ∈ X.

4 Theorem. If (C, F ) is a concrete category, then (C op , F ) is a concrete category for some (faithful) functor F : C op → Set. Proof. Let (C, F ) be a concrete category. The functor F : C → Set induces a faithful (covariant) functor F : C op → Setop (same object map and same morphism map). 3 can be regarded as a covariant functor P : Setop → Set. Claim: P is faithful. Let α, β : Y → X be morphisms in Setop and assume that α = β. Then the functions α, β : X → Y are not equal, implying that α(x) = β(x) for some x ∈ X.

Then there exists β : F (y) → F (x) such that β F (α) = 1F (x) . Since F is full, we have β = F (β) for some β : y → x. Therefore, F (βα) = F (β)F (α) = β F (α) = 1F (x) = F (1x ) and, since F is faithful, we get βα = 1x , so that α is split monic. An analogous proof shows that F reflects split monics and hence isomorphisms. This proves (ii). 3 Example (Circle is not homeomorphic to sphere) We use the preceding theorem to show that the circle S 1 is not homeomorphic to the sphere S 2 . Suppose, to the contrary, that α : S 1 → S 2 is an isomorphism in Top.