Remembering the last class: Definition: Let S be a subset of a topological space X. We say that S is compact if every open cover has a finite undercover. … This shows that an infinite set cannot be compact (in the discrete topology) since that particular cover would not have a finite cover.
Can an infinite set have a single limit?
Theorem 212 (BolzanoWeierstrass): Every bounded infinite set of real numbers has at least one limit point. Hint: It is clear that some bounded infinite sets of real numbers have no more than one limit point (e.g. the set represented by the sequence {2−n}) – think here of a sequence containing a together represents.
Can an open set be compact?
Thus the open cover C defined in (⋆) has no finite undercover, and hence (0,1) is not compact. In many topologies, open sets can be compact. In fact, the empty set is always compact. the empty set and the real series are open.
What makes a set compact?
A set S of real numbers is called compact if every sequence in S has a subsequence that converges to an element in S again.
Can a set be infinite?
An infinite set is a set whose elements cannot be counted. An infinite set is a set that has no last element. An infinite set is a set that can be brought into one-to-one correspondence with a proper subset of itself. … A set is infinite if it can be mapped to a proper subset.
Are all closed sets compact?
In every topological vector space (TVS) a compact subset is complete. However, every non-Hausdorff TVS contains compact (and therefore complete) subsets that are not closed. If A and B are disjoint compact subsets of a Hausdorff space X, then there exist disjoint open sets U and V in X with A ⊆ U and B ⊆ V.
Is the empty set compact?
Since the complement of an open set is closed, and the empty set and X are complements of each other, the empty set is also closed, making it a closed set. Also, the empty set is compact because every finite set is compact. The closure of the empty set is empty.
Is 0 a finite number?
Finite numbers are real numbers with ne = +infinity. Negative numbers cannot be finite when it comes to distances since they serve as direction. 0 neither finite nor infinite. 0 cannot be measured because it has no value, and it has no meaning because it gets nowhere.
How do you know if a set is infinite or finite?
If a set has infinitely many elements, then it is infinite, and if the elements are countable, then it is finite. 18