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King Abdul Aziz University Math 463 FAR Second Semester 1433/

1434

Faculty of Science General Topology Date: 8 /7 /1434 Mathematics Departement Final Exam Time:8- 10

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Name: Co. No. Serial number

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Answer the following:

(QI) Mark true or false and justify your answer.

1. LetX,be a topological space andA _⊂ X. ThenA^c _⊂ ext_A_.0.5

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2. x ∈ A^′ iff x ∈ A−x1.5

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3. LetAbe any subset of a topological space X, thenbA  if and only ifAis both open and closed. 2

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4. Every finiteT1−spaceX is a discrete space. 1.5

6. The identity mapi : X,1 → X,2is continuous if and only if1 ⊂ 2. 2

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7. The discrete space Xis separable if and only if it is countable. 2

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8. LetAbe a nonempty set of a metric spaceX. Thenx _∈ A iffd_x,A_{ } 0. 2

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9. Any subspace of a second countable space is second countable. 2

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10. Every indiscrete space is disconnected. 1

2

(QII) Prove the following:

1. Letf : X → Ybe a function from the topological spaceX,X into the topological spaceY,Y.

Prove the following are equivallent: 3

a. for each closed set BinY,f⁻¹_B_is closed inX, b. for each subsetB ⊂ Y,f⁻¹B ⊂ f⁻¹B

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2. Every compact Hausdorff space is normal. 5

3. LetCbe a connected subset of the space X. Then a. every setBsuch that C _⊂ B _⊂ Cis also connected, 3

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b. B is connected 1

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4. LetX,d_ be a metric space. Then the set of all open balls inX is a base for a topology onX. 4

4

5. Compactness is a topological property. 4

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6. LetAbe a closed subset of aT4 −space. Show thatAwith the relative topology is also T4−space. 4