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 E X N1- If NE N(x) then x E N N2- If NE N(x), NC M then M = N(x) N3- If NEN(x), M = N(x) then NO MEN(x) T-AEX, 4*, *S] N4- If NEN(x) then 3M EN(x) such that MCN then MEN(y) for any yЄM Show that there exist a unique topology τ on X. Q2\a\let (X,T) be the topology space and ẞET show that ẞ is base for a topology on X iff for any G open set,x E G then there exist A = ẞ such that x E A ≤ G. b\Let ẞ is a collection of open sets in X show that ẞ is base for a topology on X iff for each x EX the collection ẞx = {BEẞ \xЄ B} is is a nbhd base at x. Q31 Choose only two: al Let A be a subspace of a space X show that FSA is closed iff
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 E X N1- If NE N(x) then x E N N2- If NE N(x), NC M then M = N(x) N3- If NEN(x), M = N(x) then NO MEN(x) T-AEX, 4*, *S] N4- If NEN(x) then 3M EN(x) such that MCN then MEN(y) for any yЄM Show that there exist a unique topology τ on X. Q2\a\let (X,T) be the topology space and ẞET show that ẞ is base for a topology on X iff for any G open set,x E G then there exist A = ẞ such that x E A ≤ G. b\Let ẞ is a collection of open sets in X show that ẞ is base for a topology on X iff for each x EX the collection ẞx = {BEẞ \xЄ B} is is a nbhd base at x. Q31 Choose only two: al Let A be a subspace of a space X show that FSA is closed iff
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...
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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 E X
N1- If NE N(x) then x E N
N2- If NE N(x), NC M then M = N(x)
N3- If NEN(x), M = N(x) then NO MEN(x)
T-AEX, 4*, *S]
N4- If NEN(x) then 3M EN(x) such that MCN then MEN(y) for any
yЄM
Show that there exist a unique topology τ on X.
Q2\a\let (X,T) be the topology space and ẞET show that ẞ is base for a
topology on X iff for any G open set,x E G then there exist A = ẞ such
that x E A ≤ G.
b\Let ẞ is a collection of open sets in X show that ẞ is base for a
topology on X iff for each x EX the collection ẞx = {BEẞ \xЄ B} is is a
nbhd base at x.
Q31 Choose only two:
al Let A be a subspace of a space X show that FSA is closed iff](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef45a887-188a-4a1a-830a-b8d63580d21c%2Fb0a76d31-1b19-44c7-ad4e-f8df7534a1e3%2F15639m_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 E X
N1- If NE N(x) then x E N
N2- If NE N(x), NC M then M = N(x)
N3- If NEN(x), M = N(x) then NO MEN(x)
T-AEX, 4*, *S]
N4- If NEN(x) then 3M EN(x) such that MCN then MEN(y) for any
yЄM
Show that there exist a unique topology τ on X.
Q2\a\let (X,T) be the topology space and ẞET show that ẞ is base for a
topology on X iff for any G open set,x E G then there exist A = ẞ such
that x E A ≤ G.
b\Let ẞ is a collection of open sets in X show that ẞ is base for a
topology on X iff for each x EX the collection ẞx = {BEẞ \xЄ B} is is a
nbhd base at x.
Q31 Choose only two:
al Let A be a subspace of a space X show that FSA is closed iff
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