Difference between revisions of "Open set"

Revision as of 20:30, 24 April 2015

Here

$(X,d)$ denotes a metric space, and

$B_r(x)$ the open ball centred at

$x$ of radius

$r$

"A set

$U$ is open if it is a neighborhood to all of its points"^[1] and neighborhood is as you'd expect, "a small area around".

A set

$N$ is a neighborhood to

$a\in X$ if

$\exists\delta>0:B_\delta(a)\subset N$

That is if we can puff up any open ball about $x$ that is entirely contained in $N$

In a topological space the elements of the topology are defined to be open sets

A subset $N$ of a Topological space $(X,\mathcal{J})$ is a neighbourhood of $p$ ^[2] if:

@@ Line 19: / Line 19: @@
 * <math>\exists U\in\mathcal{J}:p\in U\wedge U\subset N</math>
 ==See also==
+* [[Relatively open|Relatively open set]]
 * [[Closed set]]
+* [[Relatively closed|Relatively closed set]]
 ==References==