TRANSCRIBE THE FOLLOWING TEXT IN DIGITAL FORMAT Let X be a set and I be the family consisting ofthe empty set and all subsets of X whose complementis countable.To prove: I is a topology of X.TProof:Step I:Here T = {A≤X: 0, A²= finite}[T₁]: We see that ETand also we know thatx² =эхетXETi.e. ₁XET.(which is finite)Step II: [T₂]: If G₁, G₂ ET2с2=) G₁₁ G₁ ₁₂² is finite [By definition]:. G₁ UG₁₂ is finite [ Union is also finite]Now By De-morgan's Law, (G, NG₂) is finiteĐ G, NG ETStep III:[T3]: If{G₁₁: AEA} be collection ofset in J, i.e. if GAET.сThen by definition GC is finite. VAEN⇒ NEG₁: AEN } is finite→ (U{G₁₁: AEA3) is finite [By De-morgan's law]⇒ U {G₁: AENGETIn 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- countableTopology that is Countable Complement topologythat's why it relates to discrete topology.Yes, with this topology X satisfy Frechet'sproperty. But X is not Hausdorff, because itis co-finite topology.

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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: 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.