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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

A set

which is a subset of X isCould 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.

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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(d)

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Hi;

Thank you

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'

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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).**176** books currently added on Goodreads

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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'

I'm not crazy, my mother had me tested.

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The open sets in are arbitrary unions of open intervals. The complement of is , a union of two open intervals; therefore its 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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Is it possible for other clopen sets to exist than phi and x?

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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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