Category:Examples of Neighborhoods

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This category contains examples of Neighborhood (Analysis).

Topology

Let $T = \struct {S, \tau}$ be a topological space.


Neighborhood of a Set

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:

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


Neighborhood of a Point

The set $A$ can be a singleton, in which case the definition is of the neighborhood of a point.


Let $z \in S$ be a point in a $S$.

Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$.


Then $N_z$ is a neighborhood of $z$.


That is:

$\exists U \in \tau: z \in U \subseteq N_z \subseteq S$

Pages in category "Examples of Neighborhoods"

The following 2 pages are in this category, out of 2 total.