# Category:Neighborhoods

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This category contains results about **Neighborhoods**.

Definitions specific to this category can be found in Definitions/Neighborhoods.

Let $A \subseteq S$ be a subset of $S$.

A **neighborhood** of $A$, which can be denoted $N_A$, is any subset of $S$ containing an open set of $T$ which itself contains $A$.

That is:

- $\exists U \in \tau: A \subseteq U \subseteq N_A \subseteq S$

## Subcategories

This category has the following 2 subcategories, out of 2 total.

### E

- Examples of Neighborhoods (2 P)

### N

- Neighborhood Bases (5 P)

## Pages in category "Neighborhoods"

The following 40 pages are in this category, out of 40 total.

### B

### C

### I

- Image of Point under Neighborhood of Diagonal is Neighborhood of Point
- Image of Subset under Neighborhood of Diagonal is Neighborhood of Subset
- Intersection of Neighborhood of Diagonal with Inverse is Neighborhood
- Intersection of Neighborhoods in Metric Space is Neighborhood
- Intersection of Neighborhoods in Topological Space is Neighborhood
- Inverse of Neighborhood of Diagonal Point is Neighborhood

### M

### N

- Neighborhood Basis in Cartesian Product under Chebyshev Distance
- Neighborhood Filter of Point is Filter
- Neighborhood iff Contains Neighborhood
- Neighborhood in Metric Space has Subset Neighborhood
- Neighborhood in Open Subspace
- Neighborhood in Topological Space has Subset Neighborhood
- Neighborhood in Topological Subspace
- Neighborhood of Diagonal induces Open Cover
- Neighborhood of Point in Metrizable Space contains Closed Neighborhood
- Neighborhoods in Standard Discrete Metric Space
- Neighbourhood of Point Contains Point of Subset iff Distance is Zero

### P

### S

- Set is Neighborhood of Subset iff Neighborhood of all Points of Subset
- Set is Open iff Neighborhood of all its Points
- Space is Neighborhood of all its Points
- Subset of Standard Discrete Metric Space is Neighborhood of Each Point
- Superset of Neighborhood in Metric Space is Neighborhood
- Superset of Neighborhood in Topological Space is Neighborhood