1.
Compact Space is studied in ......?
Correct Answer
A. Mathematics
Explanation
Compact space is a concept studied in mathematics. In topology, a compact space is a mathematical space that has the property that every open cover of the space has a finite subcover. This property is important in various areas of mathematics such as analysis, algebraic geometry, and differential geometry. Compact spaces have many interesting properties and are often used to prove theorems and solve problems in different branches of mathematics. Therefore, the correct answer for the question is mathematics.
2.
Compact Space has the following examples except?
Correct Answer
D. A triangle
Explanation
A triangle is not an example of a compact space because it is not a closed and bounded set. In order for a space to be compact, it must be closed and contain all of its limit points. A triangle does not satisfy these conditions as it has open edges and does not contain all of its limit points. On the other hand, a rectangle, a closed interval, and a finite set of points are all examples of compact spaces as they are closed and bounded sets.
3.
Which sometimes the synonym of compact space?
Correct Answer
A. Compact set
Explanation
A compact set is sometimes synonymous with a compact space because a compact set is a subset of a topological space that is closed and bounded, and a compact space is a topological space in which every open cover has a finite subcover. Both concepts refer to a space or set that is "small" in some sense, either in terms of size or in terms of the ability to cover it with a finite number of sets.
4.
In the what century did several disparate mathematical properties were understood that would later be seen as consequences of compactness?
Correct Answer
B. 19th Century
Explanation
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. This suggests that during this time, mathematicians were able to recognize the relationship between these properties and the concept of compactness, paving the way for further developments in this field.
5.
Who had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point?
Correct Answer
B. Bernard Bolzano
Explanation
Bernard Bolzano had been aware that any bounded sequence of points has a subsequence that must eventually get arbitrarily close to some other point.
6.
What year did Bernard Bolzano became aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point?
Correct Answer
C. 1817
Explanation
In 1817, Bernard Bolzano became aware that any bounded sequence of points has a subsequence that must eventually get arbitrarily close to some other point.
7.
Bernard Bolzano had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a .....?
Correct Answer
D. Limit point
Explanation
Bernard Bolzano was aware that any bounded sequence of points has a subsequence that must eventually get arbitrarily close to some other point. This point is called a limit point.
8.
Bolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into how many parts?
Correct Answer
A. Two
Explanation
Bolzano's proof relied on the method of bisection, which involves dividing an interval into two equal parts. By dividing the sequence into two parts, Bolzano was able to narrow down the interval where the root of the sequence lies. This iterative process of dividing the interval in half allows for a more precise determination of the root.
9.
The full significance of Bolzano's theorem, and its method of proof was rediscovered by .....?
Correct Answer
C. Karl Weierstrass
Explanation
Karl Weierstrass rediscovered the full significance of Bolzano's theorem and its method of proof. Weierstrass was a prominent mathematician who made significant contributions to the field of analysis. His rediscovery of Bolzano's theorem helped to establish the foundations of modern analysis and furthered our understanding of mathematical concepts.
10.
The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost how many years later when it was rediscovered by Karl Weierstrass?
Correct Answer
A. 50 years
Explanation
Bolzano's theorem was not fully appreciated and understood until it was rediscovered by Karl Weierstrass almost 50 years later. This suggests that it took a considerable amount of time for the significance and method of proof of Bolzano's theorem to be recognized and appreciated by the mathematical community.