# Category:Particular Point Topology

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This category contains results about Particular Point Topology.

Definitions specific to this category can be found in Definitions/Particular Point Topology.

Let $S$ be a set which is non-empty.

Let $p \in S$ be some **particular point** of $S$.

We define a subset $\tau_p$ of the power set $\powerset S$ as:

- $\tau_p = \set {A \subseteq S: p \in A} \cup \set \O$

that is, all the subsets of $S$ which include $p$, along with the empty set.

Then $\tau_p$ is a topology called the **particular point topology on $S$ by $p$**, or just **a particular point topology**.

## Subcategories

This category has the following 6 subcategories, out of 6 total.

### I

### L

### P

### S

## Pages in category "Particular Point Topology"

The following 45 pages are in this category, out of 45 total.

### A

### C

- Closed Set in Particular Point Space has no Limit Points
- Closure in Infinite Particular Point Space is not Compact
- Closure of Open Set of Particular Point Space
- Compact Space in Particular Point Space
- Convergent Sequence in Particular Point Space
- Countable Particular Point Space is Lindelöf
- Cover of Doubletons of Infinite Particular Point Space has no Locally Finite Refinement

### I

- Infinite Particular Point Space is not Compact
- Infinite Particular Point Space is not Countably Metacompact
- Infinite Particular Point Space is not Countably Paracompact
- Infinite Particular Point Space is not Metacompact
- Infinite Particular Point Space is not Strongly Locally Compact
- Infinite Particular Point Space is not Weakly Countably Compact
- Interior of Closed Set of Particular Point Space

### P

- Particular Point Space is First-Countable
- Particular Point Space is Irreducible
- Particular Point Space is Locally Compact
- Particular Point Space is Locally Path-Connected
- Particular Point Space is Non-Meager
- Particular Point Space is not Arc-Connected
- Particular Point Space is not Ultraconnected
- Particular Point Space is Path-Connected
- Particular Point Space is Pseudocompact
- Particular Point Space is Scattered
- Particular Point Space is Separable
- Particular Point Space is T0
- Particular Point Space is Weakly Locally Compact
- Particular Point Space less Particular Point is Discrete
- Particular Point Topology is Closed Extension Topology of Discrete Topology
- Particular Point Topology is Topology
- Particular Point Topology with Three Points is not T4
- Point in Particular Point Space is not Omega-Accumulation Point