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Countability axioms

Axioms describing the 'size' of the topology, such as first-countable and second-countable.
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The statement of the theorem

Let (X,τ)(X, \tau) be a topological space. \n\n1. **First-Countability Axiom:** XX is first-countable if for every point x∈Xx \in X, there exists a countable family of open sets textBx={Un}n∈N\\text{B}_x = \{U_n\}_{n \in \mathbb{N}} (a local basis at xx) such that for every open neighborhood VV of xx, there exists an n∈Nn \in \mathbb{N} such that Un⊆VU_n \subseteq V. \n\n2. **Second-Countability Axiom:** XX is second-countable if there exists a countable basis textB={Vn}n∈N\\text{B} = \{V_n\}_{n \in \mathbb{N}} for the topology τ\tau. This means that for every open set W∈τW \in \tau, WW can be written as a union of elements from textB\\text{B}: W=⋃n∈NVn′W = \bigcup_{n \in \mathbb{N}} V_n' where textB′={Vn′}n∈N⊆B\\text{B}' = \{V_n'\}_{n \in \mathbb{N}} \subseteq \text{B} and Vn′∈BV_n' \in \text{B}.\n\nThese axioms characterize spaces whose topological structure is sufficiently 'small' or 'well-behaved' relative to countable sets, often implying properties like separability or metrizability under additional conditions.