# Boundedness in extended complex plane

• matness
In summary, the conversation discusses the concept of boundedness in the extended complex plane, denoted as C'. The extended complex plane is not considered bounded because it is not contained within any open ball of finite radius. However, it is homeomorphic to a sphere and the real line, which are both bounded, but this does not necessarily mean that C' is bounded. The topic of compactness is also mentioned, but it is clarified that compactness does not imply boundedness. It is noted that the extended complex plane cannot be bounded without a metric being defined.
matness
what does it mean to be bounded in extended complex plane (say C')?
can we say all subsets of C' is bdd because C' itself bdd?

The extended complex plane, in my opinion, is not bounded. Bounded ought to mean, for a metric space, contained in some open ball of *finite* radius.

to be more explicitly
1)if i use spherical metric( i.e. i think on the sphere ) then C equivalently sphere without north pole is bdd .
why adding one point namely inf. changes boundedness
2)another point of view : i can prove C' is compact so it is closed and bdd

maybe iam missing somethings but i can't find them

You are missing something. Just because the extended complex plane is homeomorphic to a sphere does not mean it is bounded. The real line is homeomorphic to (-1,1), and I doubt you think that R is bounded.

Compact is not the same as closed and bounded. Cu{oo} is not a subset of R^3. It is homeomorphic to a subset of R^3, but 'size' is not invariant under homeomorphism.

i didnt say it is bdd because S^2 is in R^3. for example an open ball with center at inf. and radius some small quantiity (say eps.) contains
{ z€C s.t. |z|>eps } U {inf} by the way it is defined .
here what i stuck : why adding only one point(o.k. now it is a ball)
changes boundedness.
if you say i can't take inf. as a center i won't insist on that
but i am pretty sure i saw somewhere compactness implies bddness.

It is not bounded. Boundedness applies to subsets of metric spaces. What is the metric you're using? How far from 2 is infinity? Compact is not the same as closed and bounded. The point is you cannot have balls of infinite radii. I said nothing about infinity being or not being the centre of an open set, but as it happens, you cannot have infinity as the centre of a ball defined by the metric because there is no metric given.

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now there is a new thing that comes to my mind.
the reason that i cannot say such a thing because a set is bdd whenever that this set is a subset (proper)?.otherwise we would say C is bdd in C
OR R is bdd in R e.t.c.
still wrong?

You can't say it is bounded because you need to use a metric to define the size of your sets. You haven't put a metrc on the extended complex plane.

:) i got it with one second difference . so i stop bothering.
thanks

## What is "boundedness" in the extended complex plane?

Boundedness in the extended complex plane refers to the property of a function or set of points to remain within a certain range of values. In the extended complex plane, this range includes both real and imaginary infinity.

## How is boundedness determined in the extended complex plane?

Boundedness in the extended complex plane can be determined by analyzing the behavior of a function or set of points as the complex numbers approach infinity. If the function or set of points remains within a finite range, it is considered bounded.

## What are some examples of bounded functions in the extended complex plane?

Examples of bounded functions in the extended complex plane include polynomials, rational functions, and trigonometric functions. These functions have finite limits as the complex numbers approach infinity, making them bounded.

## What is the significance of boundedness in the extended complex plane?

Boundedness in the extended complex plane is important in understanding the behavior of functions and sets of points that involve complex numbers. It allows us to make predictions and analyze the behavior of these functions in different regions of the complex plane.

## Are there any functions or sets of points that are unbounded in the extended complex plane?

Yes, there are functions and sets of points that are unbounded in the extended complex plane. These include exponential functions, logarithmic functions, and power functions with a complex exponent. These functions have infinite limits as the complex numbers approach infinity, making them unbounded.

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