Unless f is a map from D to itself, your notation doesn't make sense. The symbol f(D) implies that D is the domain, and f(D) is the image under f. The symbol f-1(D) implies that D is now the range.analysis001 said:Homework Statement
I need to find an example of a set D[itex]\subseteq[/itex]R is compact but f-1(D) is not.
Homework Equations
f-1(D) is the pre-image of f(D), not the inverse.
analysis001 said:The Attempt at a Solution
I'm having trouble visualizing a function that would work for this scenario. Any clues would be helpful.
Mark44 said:Unless f is a map from D to itself, your notation doesn't make sense. The symbol f(D) implies that D is the domain, and f(D) is the image under f. The symbol f-1(D) implies that D is now the range.
Mark44 said:The image of the sine function is the interval [-1, 1]. The inverse image of [-1, 1] is R. If we're talking about inverse image as opposed to function inverse (f-1), this should work. If you really do mean f-1 as the function inverse, then no, it won't work, as the sine function isn't one-to-one, so doesn't have an inverse that is a function.
No. You're getting all balled up in the notation and not understanding what it's supposed to mean. The problem is to find a function and a set D (NOT f(D)) that is compact, but the inverse image of D is not compact.analysis001 said:Yes, it's talking about the inverse image, not the function inverse. I don't really see how f(D)=sin(D) would work though. If the question was to find a f(D)
analysis001 said:[itex]\subseteq[/itex]R where f(D) is compact but f-1(D) is not then I see how f(D)=sin(D) would work, because f(D)=[-1,1] is compact but f-1(D)=R is not (I think). Maybe I'm just understanding it wrong because I don't see how f(D)=sin(D) works.
A compact set is a subset of a metric space that is closed and bounded. In other words, it contains all of its limit points and can be contained within a finite distance.
While a compact set is always closed, a closed set is not always compact. A closed set can have infinitely many points and still be compact, while a closed set with infinitely many points is not always compact.
The Heine-Borel theorem states that a subset of Euclidean space is compact if and only if it is closed and bounded. This is a fundamental theorem in the study of compact sets, as it provides a necessary and sufficient condition for a set to be compact in Euclidean space.
Compact sets are used in functional analysis to study the behavior of functions on compact sets. For example, the extreme value theorem states that a continuous function on a compact set must attain its maximum and minimum values within that set.
Yes, a non-metric space can have a compact set if it satisfies the conditions of a compact set, such as being closed and bounded. However, the concept of compactness is defined differently in non-metric spaces compared to metric spaces.