Definitions Closure (topology)

1 definitions

1.1 point of closure
1.2 limit point
1.3 closure of set

definitions
point of closure

for s subset of euclidean space, x point of closure of s if every open ball centered @ x contains point of s (this point may x itself).

this definition generalises subset s of metric space x. expressed, x metric space metric d, x point of closure of s if every r > 0, there y in s such distance d(x, y) < r. (again, may have x = y.) way express x point of closure of s if distance d(x, s) := inf{d(x, s) : s in s} = 0.

this definition generalises topological spaces replacing open ball or ball neighbourhood . let s subset of topological space x. x point of closure (or adherent point) of s if every neighbourhood of x contains point of s. note definition not depend upon whether neighbourhoods required open.

limit point

the definition of point of closure closely related definition of limit point. difference between 2 definitions subtle important — namely, in definition of limit point, every neighbourhood of point x in question must contain point of set other x itself.

thus, every limit point point of closure, not every point of closure limit point. point of closure not limit point isolated point. in other words, point x isolated point of s if element of s , if there neighbourhood of x contains no other points of s other x itself.

for given set s , point x, x point of closure of s if , if x element of s or x limit point of s (or both).

closure of set

the closure of set s set of points of closure of s, is, set s of limit points. closure of s denoted cl(s), cl(s),







s
¯






{\displaystyle \scriptstyle {\bar {s}}}

or





s

−





{\displaystyle \scriptstyle s^{-}}

. closure of set has following properties.

cl(s) closed superset of s.
cl(s) intersection of closed sets containing s.
cl(s) smallest closed set containing s.
cl(s) union of s , boundary ∂(s).
a set s closed if , if s = cl(s).
if s subset of t, cl(s) subset of cl(t).
if closed set, contains s if , if contains cl(s).

sometimes second or third property above taken definition of topological closure, still make sense when applied other types of closures (see below).

in first-countable space (such metric space), cl(s) set of limits of convergent sequences of points in s. general topological space, statement remains true if 1 replaces sequence net or filter .

note these properties satisfied if closure , superset , intersection , contains/containing , smallest , closed replaced interior , subset , union , contained in , largest , , open . more on matter, see closure operator below.