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Texts: Consider the function ρ: ℝ^{2} × ℝ^{2} → ℝ given by ρ (p, q) = √{9 (x_{1} - x_{2})^{2} + 4 (y_{1} - y_{2})^{2}} where p = (x_{1}, y_{1}) and q = (x_{2}, y_{2}) ∈ ℝ^{2}. a) Show that ρ is a metric. b) Describe the balls given by the previous metric. c) Prove that the metric topology given by ρ is equivalent to the usual topology on ℝ^{2}. d) Compare the metric topology given by ρ with the lexicographic order topology on ℝ^{2}. Answer the following questions about this literal: - Which topology is finer, the metric topology or the lexicographic order topology? - Are they comparable?

Name: texts consider the function r2 r2 r given by p q 9 x1 x22 4 y1 y22 where p x1 y1 and q x2 y2 r2 a show that is a metric b describe the balls given by the previous metric c prove that 06012
Uploaded: 2023-09-13T08:18:26-08:00
Duration: 3 min 38 s
Channel: Adi S
Description: texts consider the function r2 r2 r given by p q 9 x1 x22 4 y1 y22 where p x1 y1 and q x2 y2 r2 a show that is a metric b describe the balls given by the previous metric c prove that 06012

          Texts: Consider the function ρ: ℝ^{2} × ℝ^{2} → ℝ
given by ρ (p, q) = √{9 (x_{1} - x_{2})^{2} + 4 (y_{1} - y_{2})^{2}} where p = (x_{1}, y_{1}) and q = (x_{2}, y_{2}) ∈ ℝ^{2}.
a) Show that ρ is a metric.
b) Describe the balls given by the previous metric.
c) Prove that the metric topology given by ρ is equivalent to the usual topology on ℝ^{2}.
d) Compare the metric topology given by ρ with the lexicographic order topology on ℝ^{2}.
Answer the following questions about this literal:
- Which topology is finer, the metric topology or the lexicographic order topology?
- Are they comparable?

Added by Gabriel C.

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