Example: In
, , and are clopen.]]>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.]]>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).]]>Thank you
Could you give an example of a closed set on the complex plane or the Real Number line?
]]>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)
]]>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.