Sequence

Limit point

A limit point¹ generalises convergence to a limit. Given a topological space 𝑋, a point 𝑎 ∈𝑋 is called a limit point of a sequence (𝑥𝑛)∞𝑛=1 iff every (open) neighbourhood from 𝑎 contains infinite 𝑎𝑛. #m/def/topology Note that this does not imply 𝑥𝑛 →𝑎, for examples the sequence defined by 𝑎𝑛 =( −1)𝑛 is not convergent but has limit points { −1,1}

Properties

• Limit points are points contained in the closure of every end piece
• Limit points are limits of convergent subsequences in a first-countable space

#state/tidy | #lang/en | #SemBr