FAA

Question 1

Let (𝑋, 𝜏) be a topological space. Define what it means for this space to be Hausdorff

Answer 1

The space (𝑋, 𝜏) is Hausdorff if, for all 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≠ 𝑦, there exists 𝑈, 𝑉 ∈ 𝜏 such that 𝑥 ∈ 𝑈, 𝑦 ∈ 𝑉, and 𝑈 ∩ 𝑉 = ∅.

Question 2

Define what it means for a function 𝑓 ∶ (𝑋, 𝜏_𝑋) → (𝑌 , 𝜏_𝑌 ) between topological spaces to be continuous.

Answer 2

f is continuous if for every U ∈ 𝜏_Y , f^₋₁ (U) ∈ 𝜏_X.

Question 3

Let (𝑋, 𝜏) be a topological space. Define what it means for (𝑋, 𝜏) to be compact.

Answer 3

The space (𝑋, 𝜏_𝑋) is compact if every open cover of 𝑋 has a finite subcover.

Question 4

Let (𝑋, 𝜏) be a topological space and let 𝐸 ⊆ 𝑋. Define what it means for (𝑋, 𝜏) to be connected.

Answer 4

The space (𝑋, 𝜏) is connected if ∅ and 𝑋 are the only subsets of 𝑋 which are both open and closed (i.e. which are clopen).

Question 5

Let (𝑋, 𝜏) be a topological space. Define what it means for this space to be regular

Answer 5

The space is regular if, for every closed subset 𝐸 ⊂ 𝑋 and for every 𝑥 ∈ 𝑋 ⧵ 𝐸, there exist 𝑈, 𝑉 ∈ 𝜏 such that 𝑥 ∈ 𝑈, 𝐸 ⊆ 𝑉, and 𝑈 ∩ 𝑉 = ∅.

Question 6

Let (𝑋, 𝜏) be a topological space. Define what it means for ℬ₂ ⊆ 𝒫(𝑋) to be a sub-base for 𝜏.

Answer 6

The set B₂ is a sub-base for 𝜏 if the set B’ consisting of all finite intersections of subsets in B₂, i.e.

B’ := ∩_(finite) B₂ := {X} ∪ (∩_{ⁿ k=1} V_k ⊆ X : n ∈ N ∧ V1, . . . , V_n ∈ B₂ ) , is a base for 𝜏

Question 7

Let (𝑍, 𝑑₁ ) and (𝑍, 𝑑₂ ) be two metric spaces. Define what it means for 𝑑₁ and 𝑑₂ to be uniformly equivalent.

Answer 7

The metrics are uniformly equivalent if there exist constants C₁ > 0 and C₂ > 0 such that, for all w, z ∈ Z, C₁d₁(w, z) ≤ d₂(w, z) ≤ C₂d₁(w, z).

Question 8

Let (𝑋, 𝜏) be a topological space and let 𝐸 ⊆ 𝑋. Define what a path in (𝑋, 𝜏) is.

Answer 8

A path in (𝑋, 𝜏) is a continous function 𝛾 ∶ [𝑎, 𝑏] → 𝑋, for some 𝑎, 𝑏 ∈ ℝ with 𝑎 < 𝑏.

Question 9

Let (𝑍, 𝑑₁ ) and (𝑍, 𝑑₂ ) be two metric spaces. Define what it means for 𝑑₁ and 𝑑₂ to be equivalent.

Answer 9

The metrics d₁ and d₂ are equivalent if, for all sequences (z_n) ⊆ Z and all z ∈ Z,

d₁(z_n, z) → 0 as n → ∞ if and only if d₂(z_n, z) → 0 as n → ∞.

Question 10

Let (𝑋, 𝜏) be a topological space. Define what it means for this space to be normal

Answer 10

The space is normal if, for every pair of disjoint, closed subsets 𝐸, 𝐹 ⊆ 𝑋, there exist 𝑈, 𝑉 ∈ 𝜏 such that 𝐸 ⊆ 𝑈, 𝐹 ⊆ 𝑉, and 𝑈 ∩ 𝑉 = ∅.

Question 11

Let 𝑋 be a set. Define what a topology 𝜏 on 𝑋 is.

Answer 11

A topology 𝜏 on 𝑋 is a subset of 𝒫(𝑋) (i.e. a collection, or set, of subsets of 𝑋) satisfying the following three properties:

T1 ∅ ∈ 𝜏 and 𝑋 ∈ 𝜏,

T2 if 𝑈, 𝑉 ∈ 𝜏, then 𝑈 ∩ 𝑉 ∈ 𝜏,

T3 if for all 𝑖 ∈ 𝐼 (for some index set 𝐼) 𝑈_𝑖 ∈ 𝜏, then ⋃_𝑖 ∈ 𝐼 𝑈_𝑖 ∈ 𝜏.

Question 12

Let (𝑋, 𝜏_𝑋) and (𝑌 , 𝜏_𝑌 ) be topological spaces.

Define what it means for these spaces to be homeomorphic.

Answer 12

The topological spaces (𝑋, 𝜏_𝑋) and (𝑌 , 𝜏_𝑌 ) are homeomorphic if there exists a continuous bijection 𝑓 ∶ 𝑋 → 𝑌 which has continuous inverse.

Question 13

Let 𝑋 be a set. Define what it means for 𝑑 ∶ 𝑋 × 𝑋 → [0, ∞) to be a pseudo-metric on 𝑋.

Answer 13

The function 𝑑 is a metric on 𝑋 if it satisfies the following three conditions M1’ for all 𝑥 ∈ 𝑋, 𝑑(𝑥, 𝑥) = 0. M2 for all 𝑥, 𝑦 ∈ 𝑋, 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥), M3 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝑑(𝑥, 𝑦) ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦).

Question 14

Let (𝑋, 𝑑) be a metric space and (𝑥_𝑛 )_𝑛∈ℕ a sequence in 𝑋.

Define what it means for (𝑋, 𝑑) to be complete.

Answer 14

The space (𝑋, 𝑑) is complete if every Cauchy sequence in 𝑋 converges.

Question 15

Let (𝑋, 𝑑_𝑋) and (𝑌 , 𝑑_𝑌 ) be two metric spaces. Define what it means for a function to be an isometry from 𝑋 to 𝑌 and define what it means for these spaces to be isometric.

Answer 15

The function f : X → Y is an isometry, if for all x, y ∈ X, d_Y (f(x), f(y)) = d_X(x, y).

If there exists a bijective isometry between X and Y , then the two metric spaces are isometric

Question 16

Let 𝑋 be a set. Define what it means for 𝑑 ∶ 𝑋 × 𝑋 → [0, ∞) to be a metric on 𝑋.

Answer 16

The function 𝑑 is a metric on 𝑋 if it satisfies the following three conditions M1 for all 𝑥, 𝑦 ∈ 𝑋, 𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦, M2 for all 𝑥, 𝑦 ∈ 𝑋, 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥), M3 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝑑(𝑥, 𝑦) ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦).

Question 17

Let (𝑋, 𝜏) be a topological space. Define what a finite subcover of an open cover of 𝑋 is.

Answer 17

A finite subcover of the open cover {𝑈_𝑖 }_𝑖∈𝐼 is a collection of subsets {𝑈_𝑖 }_{𝑖∈𝐼′} ⊆ {𝑈_𝑖 }𝑖∈𝐼 with 𝐼 ′ ⊆ 𝐼 finite, such that ⋃_𝑖∈𝐼′ 𝑈_𝑖 = 𝑋.

Question 18

Let (𝑋, 𝜏) be a topological space. Define what it means for a subset 𝑈 ⊆ 𝑋 to be connected.

Answer 18

A subset 𝑈 ⊆ X is connected, if (𝑈, 𝜏_𝑈 ) is connected, where 𝜏_𝑈 is the subspace topology on 𝑈 induced by 𝜏 .

Question 19

Let (𝑋, 𝜏) be a topological space. Define what it means for ℬ₁ ⊆ 𝒫(𝑋) to be a base for 𝜏.

Answer 19

The set B1 is a base for τ if the following conditions are satisfied:

(B1) B₁ ⊆ τ ,

(B2) ∀U ∈ τ , there is {V_i ∈ B₁ : i ∈ I} ⊆ B₁, such that U = U_i∈I V_i (with I some index set; I = ∅ is allowed).

Question 20

Let (𝑋, 𝜏) be a topological space and let 𝐸 ⊆ 𝑋. Define what it means for 𝐸 to be path-connected.

Answer 20

The space is path-connected if for all 𝑥, 𝑦 ∈ 𝑋 there exists a path 𝛾 ∶ [𝑎, 𝑏] → 𝑋 such that 𝛾(𝑎) = 𝑥, 𝛾(𝑏) = 𝑦, and 𝛾([𝑎, 𝑏]) ⊆ 𝐸.

Question 21

If (𝑍₁ , 𝜏) is a topological space, 𝑍₂ a set, and 𝑞 ∶ 𝑍₁ → 𝑍₂ a function, define what the quotient topology on 𝑍₂ induced by 𝑞 is.

Answer 21

The quotient topology is defined as

𝜏_q := {U ⊆ Z₂ : q⁻¹ (U) ∈ 𝜏}.

Question 22

Let (𝑋, 𝑑) be a metric space and (𝑥_𝑛 )_𝑛∈ℕ a sequence in 𝑋.

Define what it means for (𝑥_𝑛 )_𝑛∈ℕ to be a Cauchy sequence.

Answer 22

The sequence (𝑥_𝑛 ) is a Cauchy sequence if, for all 𝜀 > 0 there exists an 𝑁 ∈ ℕ such that for all 𝑛, 𝑚 ≥ 𝑁, 𝑑(𝑥_𝑛 , 𝑥_𝑚) < 𝜀.

Question 23

Let (𝑋, 𝜏) be a topological space. Define what an open cover of 𝑋 is.

Answer 23

An open cover of 𝑋 is a collection of open subsets {𝑈_𝑖 }_{𝑖 ∈ 𝐼} of 𝑋 (where 𝐼 is some index set) such that ⋃_𝑖 ∈ 𝐼 𝑈_𝑖 = 𝑋.