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Let x be non-empty of points. a topological Space having open set with finite numb cal shase wh

Let x be non-empty of points. a topological Space having open set with finite numb cal shase wh

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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x be a topological Space having non-empty open set with finite number of points. Let Let y be a topological space which is homeomorphic to x. ⇒ W a homeomorphism fix →Y. ., хо Let A = {x₁, {. xn} be a finite open Subset of x. Then f(A) = {f(x₁), ---, of (x₂)} is also finite subset of y. (by(i)) To prove that f(A) is open subset of Y. ¿) ⇒ f¹: y →x is also continuous. A is open subset of x. ⇒ (f) ²(A) = { fcx ₁), ---, f(xen)} = f(A) is open Subset of y. (: f" is continuous). (( =) having non-empty open set with finite number of points" is topological property.
Transcribed Image Text:x be a topological Space having non-empty open set with finite number of points. Let Let y be a topological space which is homeomorphic to x. ⇒ W a homeomorphism fix →Y. ., хо Let A = {x₁, {. xn} be a finite open Subset of x. Then f(A) = {f(x₁), ---, of (x₂)} is also finite subset of y. (by(i)) To prove that f(A) is open subset of Y. ¿) ⇒ f¹: y →x is also continuous. A is open subset of x. ⇒ (f) ²(A) = { fcx ₁), ---, f(xen)} = f(A) is open Subset of y. (: f" is continuous). (( =) having non-empty open set with finite number of points" is topological property.
Expert Solution
Step 1

Let X be a topological space having non-empty open set with finite number of points.

Let Y be a topological space which is homeomorphic to X.

⇒ ∃ a homeomorphism f: X → Y   ---(i)

Let A = {x1, ... , xn} be a finite open subset of X. Then f(A) = f(x1), ..., fxn is also finite subset of Y. (by (i))

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