Counter Examples in Topology

Here I have these Topological Spaces under development

• Deleted Integer Topology
• Finite Particular Point Topology
• Fortissimo Space
• Half Disc Topology
• Irrational Slope Topology
• Uncountable Finite Complement Space

So, the sources that I have used to formalize these spaces are as follows:

• Counterexamples in Topology
• [N. Bourbaki, General Topology][bourbaki1966]

The main function of this project is to make Counter Examples in Topology, So, given a conjecture in topology, the computer can specialize any of these counter-examples for this conjecture and give a quick disproof of the conjecture. Here, I have specifically focussed on implementing the counter examples in topology baased on these seperation axioms :-

• T0 axiom: If a,b ∈ X; there exist an open set O ∈ τ such that a ∈ O and b ∉ O, or b ∈ O and a ∉ 0
• T1 axiom : If a,b ∈ X , there exists open sets U and V st. x ∈ U,y ∈ V and x ∉ V,y ∉ U.
• T2 axiom : If a,b ∈ X , there exists open sets U and V st. x ∈ U,y ∈ V and U ∩ V = ∅.
• T2.5 axiom : If a,b ∈ X , there exists open sets U and V st. x ∈ U,y ∈ V and closure U ∩ closure V = ∅.
• T3 axiom: If A is a closed set and b is a point not in A, there exists open sets U and V st. A ⊆ U,b ∈ V and U ∩ V = ∅.
• T4 axiom: If A and B are disjoint sets then there exists open sets U and V st. A ⊆ U,B ⊆ V and U ∩ V = ∅.
• T5 axiom : If A and B are seperated sets i.e closure A ∩ B = ∅ and A ∩ closure B = ∅ , then there exists open sets U and V st. A ⊆ U,B ⊆ V and U ∩ V = ∅. In Bourbaki and lean's definition it also inherits the T1 Space.

So, in order to distinguish these Seperation Axioms , the plan of the project is to formalize these Topological Spaces :-

• Deleted Integer Topology : A Topological Space which is not T0
• Finite Particular Point Topology : A Topological Space which is not T1 but is T0.
• Uncountable Finite Complement Topology : A Topological Space which is not T2 but is T1.
• Irrational Slope Topology : A Topological Space which is T2.5 but not T2.
• Half Disc Topology : A Topological Space which is T3 but not T2.5
• Fortissimo Space : A Topological Space which is T5.

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