Difference between revisions of "Open set"

Revision as of 00:23, 9 March 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

@@ Line 15: / Line 15: @@
 In a [[Topological space|topological space]] the elements of the topology are defined to be open sets
+==See also==
+* [[Closed set]]
 ==References==
 {{Definition|Topology|Metric Space}}