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

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,271
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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.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
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#2 2014-01-13 21:47:13

Nehushtan
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From: London
Registered: 2013-03-09
Posts: 612
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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: The Complex Plane
Registered: 2011-01-29
Posts: 16,271
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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'
'Humanity is still kept intact. It remains within.' -Alokananda

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

Nehushtan
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From: London
Registered: 2013-03-09
Posts: 612
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: The Complex Plane
Registered: 2011-01-29
Posts: 16,271
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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'
'Humanity is still kept intact. It remains within.' -Alokananda

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

Nehushtan
Member
From: London
Registered: 2013-03-09
Posts: 612
Website

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: The Complex Plane
Registered: 2011-01-29
Posts: 16,271
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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'
'Humanity is still kept intact. It remains within.' -Alokananda

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

Nehushtan
Member
From: London
Registered: 2013-03-09
Posts: 612
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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