What does vanishing at infinity mean for a topological space?

In summary, the set of continuous functions of compact support in a locally compact Haussorff space form a normal vector space C_c(X) with the supremum norm. The completion of this space is the space C_0(X) of functions vanishing at infinity, which is defined as functions that can be continuously extended to the one-point compactification of X and have a value of 0 at infinity. However, this definition may seem abstract and not very practical. There is a more practical characterization of vanishing at infinity, which can be found on "en.wikipedia.org/wiki/Vanish_at_infinity" . To prove that this definition is equivalent to the one given above, one can consider the topology of the one-point
  • #1
lugita15
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If X is a locally compact Haussorff space, then the set of continuous functions of compact support form a normal vector space C_c(X) with the supremum norm, and the completion of this space is the space C_0(X) of functions vanishing at infinity, i.e. the space of functions f such that f can be contintuous extended to a function g on the one point compactification of X and g(∞) = 0.

However, this definition is rather abstract, and requires investigating continuous extensions on the one-point compactification, something that you wouldn't expect for a notion as simple as a limit at infinity. So is there a more practical characterization of vanishing at infinity, something that is directly in terms of X?

Any help would be greatly appreciated.

Thank You in Advance.

EDIT: I found exactly the characterization I was looking for in "en.wikipedia.org/wiki/Vanish_at_infinity" . So now the question becomes, how would you prove that the definition given in the Wikipedia article is equivalent to the one I gave above?
 
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  • #2
lugita15 said:
So now the question becomes, how would you prove that the definition given in the Wikipedia article is equivalent to the one I gave above?
This follows easily if we consider the topology for the one point compactification. What does a neigbourhood of the point at infinity look like?
 

Related to What does vanishing at infinity mean for a topological space?

1. What is the definition of vanishing at infinity for a topological space?

Vanishing at infinity refers to the behavior of a function or sequence in a topological space as its input values or indices approach infinity. This means that the function or sequence becomes arbitrarily close to 0 as its input values or indices become infinitely large.

2. How is vanishing at infinity related to the concept of limits in calculus?

Vanishing at infinity is closely related to the concept of limits in calculus. In fact, the idea of a function vanishing at infinity can be thought of as a specific type of limit, where the input values approach infinity. This type of limit is often used to describe the behavior of functions or sequences in infinite domains or as input values approach infinity.

3. What are some examples of topological spaces where vanishing at infinity is commonly studied?

Vanishing at infinity is a concept that is commonly studied in topological spaces such as Euclidean spaces, metric spaces, and compact spaces. It is also frequently studied in the context of sequences and functions on the real numbers, as well as in other infinite-dimensional spaces in mathematics and physics.

4. How is vanishing at infinity related to the concept of continuity in topology?

In topology, continuity is a fundamental concept that describes the behavior of a function or space as its input values change. Vanishing at infinity is related to continuity in that it describes the behavior of a function or sequence as its input values approach infinity. In other words, a function or sequence that vanishes at infinity is continuous at infinity.

5. Can vanishing at infinity occur in discrete spaces?

Yes, vanishing at infinity can occur in discrete spaces. In fact, in discrete spaces where the input values are integers or discrete indices, the concept of vanishing at infinity can be extended to describe the behavior of sequences or functions as the input values approach the maximum or minimum value in the discrete domain.

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