# Category:Limit Points

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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.

### A

- Accumulation Points (8 P)

### C

- Condensation Points (10 P)

### E

### L

- Limit Points of Filter Bases (1 P)

### O

- Omega-Accumulation Points (8 P)

## Pages in category "Limit Points"

The following 23 pages are in this category, out of 23 total.

### A

### E

### L

- Limit Point iff Superfilter Converges
- Limit Point in Metric Space iff Limit Point in Topological Space
- Limit Point of Sequence in Discrete Space not always Limit Point of Open Set
- Limit Point of Sequence is Accumulation Point
- Limit Point of Sequence is Adherent Point of Range
- Limit Point of Sequence may only be Adherent Point of Range
- Limit Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range
- Limit Point of Set may or may not be Element of Set
- Limit Point of Subset is Limit Point of Set
- Limit Point of Subset of Metric Space is at Zero Distance
- Local Basis Test for Limit Point