# Select each and every function, f, that is a homeomorphism from the interval (0,1) to the real numbers, i.e. f : (0,1)

1: Select each and every function, f, that is a homeomorphism from the interval (0,1) to the real numbers, i.e. f : (0,1) → ℝ (there are 3 out of 5).

f(t) =1/t + 1/(t-1) with t ∈ (0,1)

f(t) = tan(t + π/2) with t ∈ (0,1)

f(t) = 1/sin(π t) with t ∈ (0,1)

f(t) = ln(t) + t/(1–t) with t ∈ (0,1)

f(t) = ln(t/(1-t)) with t ∈ (0,1)

15: Determine which of the following collections of subsets on ℝ are bases.

Select each and every subset of ℝ that is a basis for a topology (there are 3 out of 5).

{(n,n+1] s.t. n ∈ ℤ}

{(a-1,a)∪(a,∞) s.t. a ∈ ℝ}

{(a,∞) s.t. a ∈ ℝ}

{(n,n+1] s.t. n ∈ ℝ}

{{a} s.t. a ∈ ℝ}

16: Let W = [1,2) ⋃ [4,5] ⋃ (7,8] and consider ℝ with the lower limit topology.

Select each and every number that is a limit point of W (there are 3 out of 5).

5

2

8

4

7

17: Consider the set X = {a,b,c,d} with the topology given below:

Ƭ = {X, Ø, {a,b,d}, {a,c,d}, {b,c,d}, {a,d}, {b,d}, {c,d}, {d}}

Select each and every set that is closed in (X,Ƭ) (there are 3 out of 5).

{a,b,c}

{b,c,d}

{a,c}

{b}

{a,d}

18: Consider ℝ with the topology generated from the basis set ℬ = {(–a,a) s.t. a ∈ ℝ }

Select each and every set that is a compact set on ℝ with this topology (there are 3 out of 5).

(–2,2)

(1,4]

[–3,1)

(–2,1]

[–2,2]

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