Q1\a\ Let X be an arbitrary infinite set and let the family of all subsets F of X which do not contain a particular point x, EX and the complements FC of all finite subsets F of X show that (X,T) is a topology. b\ The nbhd system N(x) at x in a topological space X has the following properties NO- N(x) for any x Є X N1- If NE N(x) then x E N N2- If NE N(x), NC M then M = N(x) N3- If NE N(x), ME N(x)then NO MEN(x) T=ACX, 4*, AC Smit] N4- If N EN(x) then 3 MEN(x) such that MCN then MEN(y) for any Show that there exist a unique topology τ on X.
Q1\a\ Let X be an arbitrary infinite set and let the family of all subsets F of X which do not contain a particular point x, EX and the complements FC of all finite subsets F of X show that (X,T) is a topology. b\ The nbhd system N(x) at x in a topological space X has the following properties NO- N(x) for any x Є X N1- If NE N(x) then x E N N2- If NE N(x), NC M then M = N(x) N3- If NE N(x), ME N(x)then NO MEN(x) T=ACX, 4*, AC Smit] N4- If N EN(x) then 3 MEN(x) such that MCN then MEN(y) for any Show that there exist a unique topology τ on X.
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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![Q1\a\ Let X be an arbitrary infinite set and let the family of all subsets
F of X which do not contain a particular point x, EX and the
complements FC of all finite subsets F of X show that (X,T) is a topology.
b\ The nbhd system N(x) at x in a topological space X has the following
properties
NO- N(x)
for any x Є X
N1- If NE N(x) then x E N
N2- If NE N(x), NC M then M = N(x)
N3- If NE N(x), ME N(x)then NO MEN(x)
T=ACX, 4*, AC Smit]
N4- If N EN(x) then 3 MEN(x) such that MCN then MEN(y) for any
Show that there exist a unique topology τ on X.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb763665b-c39b-4441-bf1b-13529786b22f%2Ff9faa434-894b-4045-825a-a69fa76b9270%2Fcyu5vfc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q1\a\ Let X be an arbitrary infinite set and let the family of all subsets
F of X which do not contain a particular point x, EX and the
complements FC of all finite subsets F of X show that (X,T) is a topology.
b\ The nbhd system N(x) at x in a topological space X has the following
properties
NO- N(x)
for any x Є X
N1- If NE N(x) then x E N
N2- If NE N(x), NC M then M = N(x)
N3- If NE N(x), ME N(x)then NO MEN(x)
T=ACX, 4*, AC Smit]
N4- If N EN(x) then 3 MEN(x) such that MCN then MEN(y) for any
Show that there exist a unique topology τ on X.
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