1. Let X = {a, b, c, d, e, f}.Determinewhether(or)noteachofthefollowingcollectionofsubsetsofXisatopologyonXτ 1 = {X, ∅, {a}, {a, f }, {b, f }, {a, b, f }}N otethat{a, f } T{b, f } = {f } / ∈ τ 1 .Hence,τ 1 isnotatopology.τ 2 =

1. Let X = {a, b, c, d, e, f}.Determinewhether(or)noteachofthefollowingcollectionofsubsetsofXisatopologyonXτ 1 = {X, ∅, {a}, {a, f }, {b, f }, {a, b, f }}N otethat{a, f } T{b, f } = {f } / ∈ τ ₁ .Hence,τ ₁ isnotatopology.τ ₂ =

{X, ∅, {a, b, f }, {a, b, d}, {a, b, d, f }}Again, notethat{a, b, f } T{a, b, d} =

{a, b} / ∈ τ ₂ .Hence,τ ₂ isnotatopology.τ ₃ = {X, ∅, {f }, {e, f }, {a, f }}N otethat{e, f } S{a, f } =

{a, e, f } / ∈ τ ₃ }.Hence,τ ₃ isnotatopology.LetX = {a, b, c, d, e, f }.W hichof thef ollowingcollectionsof subsetsof XisatopologyonX?(J ustif yyouranswers)τ ₁ = {X, ∅, {c}, {b, d, e}, {b, c, d, e}, {b}}N otatopologysince{c} S{b} = {b, c} / ∈

τ ₁ τ ₂ = {X, ∅, {a}, {b, d, e}, {a, b, d}, {a, b, d, e}} Not a topology since

{b, d, e} S{a, b, d} = {b, d} / ∈ τ ₂ τ ₃ = {X, ∅, {b}, {a, b, c}, {d, e, f }, {b, d, e, f }}ItisatopologysinceItcontainsXand∅.Anyunionof setsinτ ₃ isinτ ₃ Anyintersectionof setsinτ ₃ isinτ ₃ 2.

2. Let X = {a, b, c, d, e, f}andτisthediscretetopologyonX.W hichofthefollowingstatementsaretrue?

X ∈ τ T rue {X} ∈ τ F alse {∅} ∈ τ F alse

∅ ∈ τ T rue

∅ ∈ XF alse {∅} ∈ XF alse {a} ∈ τ T rue a ∈ τ F alse

∅ ⊆ XT rue {a} ∈ XF alse {∅} ⊆ XF alse a ∈ XT rue X ⊆ τ F alse {a} ⊆ τ F alse {X} ⊆ τ T rue a ⊆ τ F alse

Let(X, τ )beanytopologicalspace.V erif ythattheintersectionof anyf initenumberof membersof τ isamemberof τ . LetP n bethestatementthat”If {A i } ⁿ _i=1 ∈ τ T n

i=1 A i ∈ τ ”.LetS = {n ∈ N :

P _n isatruestatement}.F romthedef initionof atopology, wehave1 ∈ N.Assumethatk ∈ Si.e.P _k isatruestatementi.e.whenever{A _i } ^k _i=1 ∈ τ T k

i=1 A _i ∈ τ .Consider{A _i } ^k+1 _i=1 ∈ τ . T ^k+1 _i=1 A _i = 

T k i=1 A _i 

T A k+1 .Byassumption, wehaveB =  T k

i=1 A _i 

∈ τ .N owf romthedef initionof atopology, wegetB T A _k+1 ∈ τ sinceB, A _k+1 ∈ τ .Hence, weget T k+1

i=1 A i ∈ τ whenever{A i } ^k+1 _i=1 ∈ τ .Hence,P k+1 isatruestatement.Hence,k+

1 ∈ S.Sobyprincipleof mathematicalinductionwehave1 ∈ S, andk + 1 ∈

Swheneverk ∈ S.Hence,S = N.Hence, theintersectionof anyf initenumberof membersof τ isamemberof τ . 3.

3. Let Rbethesetofallrealnumbers.P rovethateachofthefollowingcollectionsofsubsetsofRisatopology 4. τ 1 consistsof R, ∅andeveryinterval(−n, n), f oranypositiveinteger

Clearly,R, ∅ ∈ τ 1 F irstnotethatτ ₁ isacountablesetandhenceallweareinterestedinisinf inite(or)countablyinf initeunions.LetA _k = (−k, k).N otethatA _k 'saremonotoneincreasingsequenceofsets.Hence, anyfiniteunionoftheformS ^m _l=1 A _k

=

1

A _p wherep = max(k ₁ , k ₂ , . . . , k _m ).AndA _p ∈ τ ₁ andhenceanyf initeunionagainbelongstoτ ₁ .Anyinf initeunion S ∞

l=1 A _k

wherek _l ∈ Ncanberewritteneitherasaf initeunionof thef orm S m

l=1 A _k

(or)aninf initeunion S ∞

l=1 A _k

wherek _i 6=

k _j .T heclaimnowisthatanyinf initeunionwherek _i 6= k _j isRi.e. S ∞

l=1 A _k

= Rwherek i 6= k _j .T oprovethis, weprovethetwowayinclusion.N otethatA _k

∈ R.Hence, S ∞

l=1 A _k

⊆ R.F urthergivenanyx ∈ R, byarchimedianproperty∃n ∈ Nsuchthatx ∈ A n .F urthersinceitisaninf initeunion,∃k _l suchthatx ∈

A _n ⊆ A _k

.Hence, S ∞

l=1 A _k

⊇ R.F urther,A n

S R = R ∈ τ 1 andA _n S ∅ =

A _n ∈ τ ₁ .Hence, anyunionisalsoinτ ₁ .N owweneedtoprovethelastclaimthatintersectionof anytwoelementsof τ ₁ givesanelementinτ ₁ .ConsiderA _m , A _n ∈ τ ₁ .Bywell−orderingprinciple, wegetmn (or) m = n(or)mn.Ifm =

n, thenA _m T A _n = A _n ∈ τ ₁ .If mn, thenA _m T A _n = A _m ∈ τ ₁ .If mn, thenA _m T A _n = A _n ∈ τ ₁ .F urther,A _n T

R = A n ∈ τ ₁ andA _n T ∅ = ∅ ∈ τ 1 .Hence, intersectionof anytwoelementsof τ ₁ givesanelementinτ ₁ .τ ₂ consistsof R, ∅andeveryinterval[−n, n], f oranypositiveintegerClearly,R, ∅ ∈ τ ₂

F irstnotethatτ ₂ isacountablesetandhenceallweareinterestedinisinf inite(or)countablyinf initeunions.LetA _k = [−k, k].N otethatA _k 'saremonotoneincreasingsequenceofsets.Hence, anyfiniteunionoftheformS ^m _l=1 A _k

= A _p wherep = max(k ₁ , k ₂ , . . . , k _m ).AndA _p ∈ τ ₂ andhenceanyf initeunionagainbelongstoτ ₂ .Anyinf initeunion S ∞

l=1 A _k

wherek _l ∈ Ncanberewritteneitherasaf initeunionof thef orm S m

l=1 A _k

(or)aninf initeunion S ∞

l=1 A _k

wherek _i 6=

k _j .T heclaimnowisthatanyinf initeunionwherek _i 6= k _j isRi.e. S ∞

l=1 A _k

= Rwherek i 6= k _j .T oprovethis, weprovethetwowayinclusion.N otethatA _k

∈ R.Hence, S ∞

l=1 A _k

⊆ R.F urthergivenanyx ∈ R, byarchimedianproperty∃n ∈ Nsuchthatx ∈ A n .F urthersinceitisaninf initeunion,∃k _l suchthatx ∈

A _n ⊆ A _k

.Hence, S ∞

l=1 A _k

⊇ R.F urther,A n

S R = R ∈ τ 2 andA _n S ∅ =

A _n ∈ τ ₂ .Hence, anyunionisalsoinτ ₂ .N owweneedtoprovethelastclaimthatintersectionof anytwoelementsof τ ₂ givesanelementinτ ₂ .ConsiderA _m , A _n ∈ τ ₂ .Bywell−orderingprinciple, wegetmn(or)m = n(or)mn.If m =

n, thenA _m T A _n = A _n ∈ τ ₂ .If mn, thenA _m T A _n = A _m ∈ τ ₂ .If mn, thenA _m T A _n = A _n ∈ τ ₂ .F urther,A _n T

R = A n ∈ τ ₂ andA _n T ∅ = ∅ ∈ τ ₂ .Hence, intersectionof anytwoelementsof τ ₂ givesanelementinτ ₂ . 5.

5. τ 3 consistsof R, ∅andeveryinterval[n, ∞), f oranypositiveintegerClearly,R, ∅ ∈

τ ₃ F irstnotethatτ ₃ isacountablesetandhenceallweareinterestedinisinf inite(or)countablyinf initeunions.LetA _k = [k, ∞).N otethatA _k 'saremonotonedecreasingsequenceofsets.Hence, anyfiniteunionoftheformS ^m _l=1 A _k

=

A _p wherep = min(k ₁ , k ₂ , . . . , k _m ).AndA _p ∈ τ ₃ andhenceanyf initeunionagainbelongstoτ ₃ .Anyinf initeunion S ∞

l=1 A _k

wherek _l ∈ NequalsA p wherep = min(k ₁ , k ₂ , . . .).F urther,A _n S

R = R ∈ τ 3 andA _n S ∅ =

A _n ∈ τ ₃ .Hence, anyunionisalsoinτ ₃ .N owweneedtoprovethelastclaimthatintersectionof anytwoelementsof τ ₃ givesanelementinτ ₃ .ConsiderA _m , A _n ∈ τ ₃ .Bywell−orderingprinciple, wegetmn(or)m = n(or)mn.If m = n, thenA _m T A n =

A _n ∈ τ ₃ .If mn, thenA _m T A _n = A _n ∈ τ ₃ .If mn, thenA _m T A _n = A _m ∈ τ ₃ .F urther,A _n T

R = A n ∈ τ ₃ andA _n T ∅ = ∅ ∈ τ ₃ .Hence, intersectionof anytwoelementsof τ ₃ givesanelementinτ ₃ . Let Nbethesetofallpositiveintegers.P rovethateachofthefollowingcollectionsofsubsetsofNisatopology.

1. τ 1 consistsof N, ∅andeverysetof thef orm{1, 2, . . . , n}f oranypositiveintegern.T hisiscalledtheinitialsegmenttopology.Argumentsimilartotheabovequestionτ 2 consistsof N, ∅andeverysetof thef orm{n, n+

1, . . .}f oranypositiveintegern.T hisiscalledthef inalsegmenttopology.Argumentsimilartotheabovequestion

2

2. Let XbeaninfinitesetandτbeatopologyonX.IfeveryinfinitesubsetofXisinτ, provethatτisthediscretetopology.

T heideaistoprovethateverysingletonsetisinthetopology.Byapropositionprovedearlier, itf ollowsthatτ isthediscretetopology.T oprovethis, f irstobservethatgivenaninf initeset,X, wecanchooseaninf initesetAsuchthatX\Aisalsoinf inite.F orinstance, if thegivensetXiscountablyinf inite, thenwecansetabijectionf romNtothesetXi.e.wecanwritethesetXas{x k :

k ∈ N}.LetA = {x k : k ∈ Z ⁺ odd }.Clearly, bothAandX\Aarecountablyinf initesubsetsof X.If thegivensetXisuncountablyinf inite, thenbyaxiomof choice, itispossibletochooseacountablyinf initesubset, sayAof X.Hence,X\Aisalsoaninf initeset, inf actitisuncountablyinf inite.(Iwasinitiallyunsureif thiscouldbedone.Butthankstomath.stackexchange, ArturoM agidintoldmethatthiswaspossiblewiththe ¨ Axiomof choice”.)N owwehaveAandX\Atobeinf initesets.F oranyx ∈ X, considerA∪{x}andX\A∪{x}.T hesetwoareinf initesetsandhenceboththesetsareinthetopologyτ andhencetheintersectionof thetwosetsmustalsobeinthetopologyτ .T heintersectionof theabovetwosetsisnothingbut{x}.T hisistruef oreveryset{x}.Hence, everysingletonsetbelongstothetopologyandhencethetopologyisdiscrete.

Let Rbethesetofallrealnumbers.P reciselythreeofthefollowingcollectionsofsubsetsofRaretopologies?IdentifytheseandjustifyyouranswerτconsistsofR, ∅andeveryinterval(a, b), foraandbanyrealnumberswithabNotatopologysinceforinstance(1, 2), (3, 4) ∈ τ but(1, 2) S(3, 4) / ∈ τ τ consistsof R, ∅andeveryinterval(−r, r) for r ∈ RIsatopologybyargumentsimilartoquestion6τ consistsofR, ∅andeveryinterval(−r, r)forr ∈

Q ⁺ N otatopologysinceif wetakerationals{x _n } ^∞ _n=1 suchthatx _n+1 = _2x ^2x

n

+1 withx ₁ =

1

2 andletA n = (−x n , x n )andconsider S ∞

k=1 A k .N otethat{x k } ^∞ _k=1 isanincreasingsequenceboundedaboveandconvergesto ^{√ 1} ₂ .T hismeansthatthesetsA k aremonotoneincreasingsequenceof sets, andA k ∈ τ . S ^∞ _k=1 A _k = 

− ^{√ 1}

2 , ^{√ 1}

2



∈ τ τ consistsof R, ∅andeveryinterval[−r, r]forr ∈ /

Q ⁺ N otatopologysinceif weconsider{x _n } ^∞ _n=1 wherex _n = 1− _n ¹ .LetA _n = [−x _n , x _n ].Clearly,A _k ∈ τ andA _k f ormamonotoneincreasingsequenceof sets. S ∞

k=1 A _k = (−1, 1) / ∈ τ

τ consistsof R, ∅andeveryinterval(−r, r)f orr ∈ R ⁺ \QNotatopologysinceifweconsider{x n } ^∞ _n=1 wherex _n = 1 −

√ 2

2n .Clearly,x _n ∈ R ⁺ \Q, ∀n ∈ N.LetA n = (−x _n , x _n ).Clearly,A _k ∈ τ andA _k f ormamonotoneincreasingsequenceof sets. S ∞

k=1 A _k = (−1, 1) / ∈ τ τ consistsof R, ∅andeveryinterval[−r, r]forr ∈ R ⁺ \QNotatopologysinceifweconsider{x n } ^∞ _n=1 where x n = √

2− _n ¹ .Clearly,x _n ∈

R ⁺ \Q,∀n ∈ N.LetA n = [−x _n , x _n ].Clearly,A _k ∈ τ andA _k f ormamonotoneincreasingsequenceof sets. S ∞

k=1 A _k =

− √ 2, √

2
/ ∈ τ τ consistsof R, ∅andeveryinterval[−r, r) for r ∈ R ⁺ N otatopologysinceif weconsider{x _n } ^∞ _n=1 wherex _n = 1− _n ¹ .Clearly,x _n ∈ R ⁺ , ∀n ∈ N.LetA n = [−x _n , x _n ).Clearly,A _k ∈ τ andA _k f ormamonotoneincreasingsequenceof sets. S ∞

k=1 A _k = (−1, 1) / ∈ τ τ consistsof R, ∅andeveryinterval(−r, r]forr ∈ R ⁺ N otatopologysinceif weconsider{x _n } ^∞ _n=1 wherex _n =

1− _n ¹ .Clearly,x _n ∈ R ⁺ ,∀n ∈ N.LetA n = (−x _n , x _n ].Clearly,A _k ∈ τ andA _k f ormamonotoneincreasingsequenceof sets. S ∞

k=1 A _k = (−1, 1) / ∈ τ τ consistsof R, ∅andeveryinterval[−r, r]andeveryinterval(−r, r)forr ∈

R ⁺ IsatopologyClearly,X, ∅ ∈ τ LetA ⁽⁰⁾ r = (−r, r)andA ⁽¹⁾ r = [−r, r].ConsiderA =

 S

x∈Γ

A ⁽⁰⁾ x

 S  S

y∈Γ

A ⁽¹⁾ y



.T henA = A ⁽⁰⁾ x f orsomex ∈ R ⁺ (or)A = A ⁽¹⁾ y f orsomey ∈ R ⁺ (or)A = R.Intersectionof anytwosetsisclosedinτ τ consistsof R, ∅, every

interval [−n, n], and every interval (−r, r), forn ∈ Z ⁺ andr ∈ R ⁺ Isatopology.Againcheckthedef initions.

3