Category:Limit Points

From ProofWiki
Jump to navigation Jump to search

This category contains results about Limit Points in the context of Topology.
Definitions specific to this category can be found in Definitions/Limit Points.


A point $x \in S$ is a limit point of $A$ if and only if every open neighborhood $U$ of $x$ satisfies:

$A \cap \paren {U \setminus \set x} \ne \O$

That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.


More symbolically, a point $x \in S$ is a limit point of $A$ if and only if

$\forall U\in \tau :x\in U \implies A \cap \paren {U \setminus \set x} \ne \O\text{.}$

Subcategories

This category has the following 7 subcategories, out of 7 total.