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#1 2014-01-13 20:54:24

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 21,744
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Examples of Open and Close Sets

If

is a collection of subsets of
such that:
(1) the union of any collection of sets that are elements of
belongs to
;
(2) the intersection of any finite collection of sets that are elements of
belong to
;
(3)the empty set and
belong to
.
Then, elements of
are open sets of the topological space
.

A set

which is a subset of X is closed in the space
if its complement
is open (i.e., X \ F ∈ Ω).

Could someone give me examples of
a) Closed Sets
b) Open Sets
c) sets which are both open and closed;
d) sets which are neither closed nor open.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
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#2 2014-01-13 21:47:13

Nehushtan
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Registered: 2013-03-09
Posts: 626
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Re: Examples of Open and Close Sets

First you need an example of a topological space.

Let

and
.

Then

(a)

(its complement in
is
)

(b)

(or any member of
)

(c)

and
(these are always both open and closed in any topological space; they are called clopen sets)

(d)


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#3 2014-01-13 22:09:19

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 21,744
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Re: Examples of Open and Close Sets

Hi;

Thank you smile

Could you give an example of a closed set on the complex plane or the Real Number line?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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#4 2014-01-13 22:32:42

Nehushtan
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Registered: 2013-03-09
Posts: 626
Website

Re: Examples of Open and Close Sets

In
, any closed interval
is a closed set. This includes singleton sets as
.

In

, an example of a closed set would be the interior and boundary of a circle (or square, or polygon, or any simply connected plane figure).


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#5 2014-01-14 02:39:02

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 21,744
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Re: Examples of Open and Close Sets

Why is it a closed set? What is the compliment of a closed interval? When on R, what exactly is the topological space?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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#6 2014-01-14 06:49:53

Nehushtan
Member
Registered: 2013-03-09
Posts: 626
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Re: Examples of Open and Close Sets


The open sets in
are arbitrary unions of open intervals. The complement of
is
, a union of two open intervals; therefore it’s open (so
is closed).

The open sets in

are more complicated to describe. Basically think of an “open figure” as a connected region of the complex plane that does not include the boundary. For example, the circle
, which does not include points on the circumference. (Note that such a region need not be bounded; e.g. the half plane
is an “open figure”.) Then the open sets in
are arbitrary unions of “open figures”.

All this may be very confusing for you, I know, but this is what

and
are as topological spaces. It would be much simpler to treat them as metric spaces instead.

Last edited by Nehushtan (2014-01-14 07:03:41)


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#7 2014-01-15 05:13:20

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 21,744
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Re: Examples of Open and Close Sets

Is it possible for other clopen sets to exist than phi and x?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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#8 2014-01-15 05:22:49

Nehushtan
Member
Registered: 2013-03-09
Posts: 626
Website

Re: Examples of Open and Close Sets

Yes. In a discrete topological space
, that is one in which
is the power set of
, every subset of
is clopen.

Example: In

,
,
and
are clopen.

Last edited by Nehushtan (2014-01-15 05:26:47)


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