Let X be a set and I be the family consisting of the empty set and all subsets of X whose complement is countable. To prove: Tis a topology of X. Proof: Step I: Here T = EACX: 0, AC= finite} [T₁]: We see that ET and also we know that с x= эхет i.е. ф,хет. [ep II: [T₂]: (which is finite) If G₁, G₂ ET 2 с с ⇒) G₁₁ G₁₂² is finite [By definition] :. G₁₁ UG₁₂ is finite [ Union is also finite] Now By De-morgan's Law, (G₁, MG₂) is finite Đ GiNG ET Step III: [T3]: If {GX: AE^} be collection of set in J, i.e. if GAET. Then by definition GC is finite. AEN ⇒ NEG₁: AEN } is finite → (UEG₁₁: AEA3) is finite [By De-morgan's law] =) UE Gia: делует In this way we see that [T₁], [₂], [T₂] holds. Consequentely T is a topology of X. which is known as Finite - Complement topology If X is countable it is related to Co- countable Topology that is Countable Complement topology that's why it relates to discrete topology. Yes, with this topology X satisfy Frechet's property. But X is not Hausdorff, because it is co-finite topology.
Let X be a set and I be the family consisting of the empty set and all subsets of X whose complement is countable. To prove: Tis a topology of X. Proof: Step I: Here T = EACX: 0, AC= finite} [T₁]: We see that ET and also we know that с x= эхет i.е. ф,хет. [ep II: [T₂]: (which is finite) If G₁, G₂ ET 2 с с ⇒) G₁₁ G₁₂² is finite [By definition] :. G₁₁ UG₁₂ is finite [ Union is also finite] Now By De-morgan's Law, (G₁, MG₂) is finite Đ GiNG ET Step III: [T3]: If {GX: AE^} be collection of set in J, i.e. if GAET. Then by definition GC is finite. AEN ⇒ NEG₁: AEN } is finite → (UEG₁₁: AEA3) is finite [By De-morgan's law] =) UE Gia: делует In this way we see that [T₁], [₂], [T₂] holds. Consequentely T is a topology of X. which is known as Finite - Complement topology If X is countable it is related to Co- countable Topology that is Countable Complement topology that's why it relates to discrete topology. Yes, with this topology X satisfy Frechet's property. But X is not Hausdorff, because it is co-finite topology.
Advanced Engineering Mathematics
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ISBN:9780470458365
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![Let X be a set and I be the family consisting of
the empty set and all subsets of X whose complement
is countable.
To prove: I is a topology of X.
T
Proof:
Step I:
Here T = {A≤X: 0, A²= finite}
[T₁]: We see that ET
and also we know that
x² =
эхет
XET
i.e. ₁XET.
(which is finite)
Step II: [T₂]: If G₁, G₂ ET
2
с
2
=) G₁₁ G₁ ₁₂² is finite [By definition]
:. G₁ UG₁₂ is finite [ Union is also finite]
Now By De-morgan's Law, (G, NG₂) is finite
Đ G, NG ET
Step III:
[T3]: If
{G₁₁: AEA} be collection of
set in J, i.e. if GAET.
с
Then by definition GC is finite. VAEN
⇒ NEG₁: AEN } is finite
→ (U{G₁₁: AEA3) is finite [By De-morgan's law]
⇒ U {G₁: AENGET
In this way we see that [T₁], [T₂], [T3] holds.
Consequentely T is a topology of X.
which is known as Finite - Complement topology.
If X is countable it is related to Co- countable
Topology that is Countable Complement topology
that's why it relates to discrete topology.
Yes, with this topology X satisfy Frechet's
property. But X is not Hausdorff, because it
is co-finite topology.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59e64be3-c4a1-45fd-a5e6-8fcac7fe7d99%2F6d72041f-d008-46a4-ac1f-ba924245b948%2Fln07rha_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X be a set and I be the family consisting of
the empty set and all subsets of X whose complement
is countable.
To prove: I is a topology of X.
T
Proof:
Step I:
Here T = {A≤X: 0, A²= finite}
[T₁]: We see that ET
and also we know that
x² =
эхет
XET
i.e. ₁XET.
(which is finite)
Step II: [T₂]: If G₁, G₂ ET
2
с
2
=) G₁₁ G₁ ₁₂² is finite [By definition]
:. G₁ UG₁₂ is finite [ Union is also finite]
Now By De-morgan's Law, (G, NG₂) is finite
Đ G, NG ET
Step III:
[T3]: If
{G₁₁: AEA} be collection of
set in J, i.e. if GAET.
с
Then by definition GC is finite. VAEN
⇒ NEG₁: AEN } is finite
→ (U{G₁₁: AEA3) is finite [By De-morgan's law]
⇒ U {G₁: AENGET
In this way we see that [T₁], [T₂], [T3] holds.
Consequentely T is a topology of X.
which is known as Finite - Complement topology.
If X is countable it is related to Co- countable
Topology that is Countable Complement topology
that's why it relates to discrete topology.
Yes, with this topology X satisfy Frechet's
property. But X is not Hausdorff, because it
is co-finite topology.
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