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A topological space is metrizable if its topology can be induced by a metric. Spaces that satisfy the second countability axiom are often metrizable.

A topological space satisfies the countable chain condition if every disjoint collection of open sets is countable. In other contexts, it's known as the Suslin's condition.

A T2 space, or Hausdorff space, is a topological space where for any two distinct points, there exist disjoint open sets containing each of the points. This ensures that limits of sequences are unique if they exist.

A T1 space is a topological space where for each pair of distinct points, there exist open sets separating them. This means each point is a closed set.

A topological space is called separable if it contains a countable dense subset. This means that there is a countable set which is dense in the space.

A topological space satisfies the first countability axiom if each point has a countable basis at that point. Implications include the fact that sequences are enough to characterize continuity and convergence.

A topological space satisfies the second countability axiom if it has a countable base for its topology. This implies the space is separable, as it contains a countable dense subset.