types of limit points

Let $X$ be a topological space and $A\\subset X$ be a subset.

A point $x\\in X$ is an $\\omega$ -accumulation point of $A$ if every open set in $X$ that contains $x$ also contains infinitely many points of $A$ .

A point $x\\in X$ is a condensation point of $A$ if every open set in $X$ that contains $x$ also contains uncountably many points of $A$ .

If $X$ is in addition a metric space, then a cluster point of a sequence $\\{x_{n}\\}$ is a point $x\\in X$ such that every $\\epsilon>0$ , there are infinitely many point $x_{n}$ such that $d(x,x_{n})<\\epsilon$ .

These are all clearly examples of limit points.

单词	TypesOfLimitPoints
释义	types of limit points Let $X$ be a topological space and $A\\subset X$ be a subset. A point $x\\in X$ is an $\\omega$ -accumulation point of $A$ if every open set in $X$ that contains $x$ also contains infinitely many points of $A$ . A point $x\\in X$ is a condensation point of $A$ if every open set in $X$ that contains $x$ also contains uncountably many points of $A$ . If $X$ is in addition a metric space, then a cluster point of a sequence $\\{x_{n}\\}$ is a point $x\\in X$ such that every $\\epsilon>0$ , there are infinitely many point $x_{n}$ such that $d(x,x_{n})<\\epsilon$ . These are all clearly examples of limit points.
随便看	NesbittsInequality Nesbitt’s inequality NestedIdealsInVonNeumannRegularRing nested ideals in von Neumann regular ring NestedIntervalTheorem nested interval theorem NestedSphereTheorem nested sphere theorem Net net NetsAndClosuresOfSubspaces nets and closures of subspaces NeumannProblem Neumann problem NeumannSeries Neumann series NeumannSeriesInBanachAlgebras Neumann series in Banach algebras NeutralGeometry neutral geometry NeutrallyStableFixedPoint neutrally stable fixed point NeutrosophicLogic neutrosophic logic NeutrosophicProbability