symv said:I am considering the set of all bounded signals given by
[tex] X = \left\{ x:\ |x(t)| \leq X_{\max}, \forall t \right\}.[/tex]
Is this set compact? Can anyone help me?
Thank you guys
g_edgar said:in what topology, in what space of functions ?
The compactness of a space refers to a mathematical property of a topological space that describes its ability to be covered by a finite number of open sets. In simpler terms, a compact space is one in which every open cover has a finite subcover.
While both compactness and connectedness are topological properties of a space, they are different concepts. Connectedness refers to the idea that a space cannot be divided into two disjoint open sets, while compactness refers to the space's ability to be covered by a finite number of open sets.
Some examples of compact spaces include a closed interval [a,b] on the real number line, a finite set with the discrete topology, and the Cantor set. Additionally, any finite topological space is also compact.
Compactness is closely related to continuity in mathematics. In particular, a function is continuous if and only if the preimage of a compact set is also compact. This means that continuity can be defined in terms of compactness.
No, not every compact space is Hausdorff. The Hausdorff property requires that for any two distinct points in a space, there exist disjoint open sets containing each point. While all Hausdorff spaces are also compact, there are examples of compact spaces that are not Hausdorff, such as the line with two origins.