4) Continuous Functions and Topological Equivalence

Question 1

Describe the condition for a function f:X→Y between two metric spaces (X,d) and (Y,d) to be continuous at a point x0 in X

Answer 1

Question 2

What is the criteria for a function to be continuous at a point in terms of sequences

Answer 2

Question 3

If(x) = 2x, show that for for any convergent sequence (wn)n≥1 in R with limit w, the sequence (f(wn))n≥1 converges to f(w)

Answer 3

Question 4

What is the Inverse Image or Pre Image of a Function

Answer 4

Question 5

Under what condition involving an open subset and the inverse image is a function continuous

Answer 5

Question 6

Show that for any set U open in (R, d), f−1(U) is open

Answer 6

Question 7

Under what condition involving a closed subset and the inverse image is a function continuous

Answer 7

Question 8

If f : X → Y and g : Y → Z are continuous functions defined on metric spaces (X, dX), (Y, dY ), and (Z, dZ), what do you know about the composition g ◦ f : X → Z

Answer 8

It is also continuous

Question 9

When is a bijection a Lipschitz equivalence

Answer 9

Question 10

If f : (X, d1) → (Y, e1) is continuous and d1 and d2 are
Lipschitz equivalent metrics and e1 and e2 are also Lipschitz equivalent. Prove that f : (X, d2) → (Y, e2) is continuous

Answer 10

Question 11

If f : (X, dX) → (Y, dY ) is a Lipschitz equivalence what does that mean about f and f^-1

Answer 11

Both f and f^−1 are continuous

Question 12

What is a Homeomorphism

Answer 12

A bijection f : X → Y is a homeomorphism whenever f and f ^−1 are both continuous.
We say (X, d) and (Y, e) are homeomorphic if there exists a homeomorphism between them

Question 13

What is the relationship between, isometry, lipschitz equivalence and homeomorphism

Answer 13

Isometry ⇒ Lipschitz equivalence ⇒ Homeomorphism

Question 14

Describe the conditions under which the identity map 1X :(X,d)→(X,e) is an isometry and a Lipschitz equivalence between two metric spaces

Answer 14

• Isometry: If and only if the metrics d and e are equal
• Lipschitz Equivalence: If and only if the metrics d and e are Lipschitz equivalent

Question 15

What does it mean for two metrics to be topologically equivalent

Answer 15

Two metrics are topologically equivalent whenever the identity function is a homeomorphism

Question 16

Under what condition involving open sets would two metrics be topologically equivalent

Answer 16

Two metrics are topologically equivalent if and only if they give rise to precisely the same open sets

Question 17

What does it mean for a metric space to be path-connected

Answer 17

A metric space X is path-connected if every two
points x0, x1 ∈ X admit a continuous function σ : [0, 1] → X such that σ(0) = x0 and σ(1) = x1.
Then σ is a path from x0 to x1 in X.

Question 18

If f : X → Y is a homeomorphism under what condition is X path-connected

Answer 18

If f : X → Y is a homeomorphism, then X is path-connected if and only if Y is path-connected

Question 19

Answer 19