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Properties

Properties of topological spaces such as compactness, connectedness, and separation.
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The statement of the theorem

Let (X,τ)(X, \tau) be a topological space. A topological property PP is a condition that can be formulated as a statement about the open sets τ\tau or the underlying set XX. Formally, we define the set of all topological properties Prop\text{Prop} as the collection of predicates P:Top→BoolP: \text{Top} \to \text{Bool} such that P(X,τ)P(X, \tau) is true if and only if the space (X,τ)(X, \tau) satisfies the condition. For instance, the property of compactness, denoted Comp\text{Comp}, is defined by the condition that every open cover {U}i∈I\{U\}_{i \in I} of XX has a finite subcover. This can be expressed axiomatically as:\n\nComp(X,τ)≜∀U⊆τ,((⋃U∈UU=X)  ⟹  ∃F⊆U such that F is finite and ⋃U∈FU=X)\text{Comp}(X, \tau) \triangleq \forall \mathcal{U} \subseteq \tau, \left( \left(\bigcup_{U \in \mathcal{U}} U = X \right) \implies \exists \mathcal{F} \subseteq \mathcal{U} \text{ such that } \mathcal{F} \text{ is finite and } \bigcup_{U \in \mathcal{F}} U = X \right) \n\nSimilarly, the property of Hausdorff separation (T2_2) requires that for any two distinct points x,y∈Xx, y \in X, there exist disjoint open neighborhoods UU and VV such that x∈Ux \in U and y∈Vy \in V. This is formalized as:\n\nT2(X,τ)≜∀x,y∈X,(x≠y  ⟹  ∃U,V∈τ such that x∈U,y∈V, and U∩V=∅)\text{T}_2(X, \tau) \triangleq \forall x, y \in X, \left( x \neq y \implies \exists U, V \in \tau \text{ such that } x \in U, y \in V, \text{ and } U \cap V = \emptyset \right) \n\nThus, the set of properties Prop\text{Prop} is the set of all such predicates that characterize specific topological structures or constraints on the open sets τ\tau.