Open Set

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于 2017-08-23 22:16:31 首次发布

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Open Set

Open sets can be formalized with various degrees of generality[1]. From the point of view of topology, the open sets can be almost anything. Geometrically, open sets have a little bit of space around each point[2]. In general topological spaces,
let $X$ be a set, and $τ\tau$ be a family of sets. $τ\tau$ is a topology on $X$ if:

$X$ and $∅\emptyset$ are in $τ\tau$ : $X \in \tau, and ~ \emptyset \in \tau $
any union of sets in $τ\tau$ is in $τ\tau$
any finite intersection of sets in $τ\tau$ is in $τ\tau$

the sets in $τ\tau$ are open sets. It means, the openness property of a set is preserved through the process of union and finite intersection.

[4] “The open set of the usual topology on $E^{n}$ are characterized as follows. A set $U$ is open if given $\in U$ we can always find a positive real number $ϵ\epsilon$ such that the ball with center $x$ and radius $ϵ\epsilon$ lies entirely in $U$ .”

Close Set

Closed set has three different definitions[4]. Each of these defitions is based on different concepts:

A subset of a topological space is closed if its complement is open.
A set is closed if and only if it contains all its limit points.
A set is closed if and only if it equal to its closure.

Closed sets contain all their boundary points, open sets contain none of theirs. Open sets have "enough space" that there is a ball around every point.

Closed sets can be so sparse they contain no metric balls at all. Differentiation is usually defined only for function defined on an open set.

[2] "Intuitively, closed set contains points such that if you are on one of them there is some direction such that if you move towards this direction by a so small movement (so small as you want) you Will fall down outside the set. However, for an open set, if you are on any of its points and you take any direction you can always move by a so small movement such you remains inside the set during the movement as well as on the destination point!"

References

[1]https://en.wikipedia.org/wiki/Open_set
[2]https://math.stackexchange.com/questions/1295138/what-is-the-mathematical-distinction-between-closed-and-open-sets
[3] Walter Dudin. Principles of Mathematical Analysis. 3rd Edition, McGraw-Hill, Inc. 1953.
[4] M. A. Armstrong. Basic Topology.2000, Springer.
[5] https://www.encyclopediaofmath.org/index.php/Covering_(of_a_\set)

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