WebDec 19, 2014 · My guess is that it is indeed an example of closed and bounded does not imply compact. Every element is less than or equal to 1, and it is closed as a whole set. … WebAug 1, 2024 · Solution 1 You are right. For a subset $A$ of metric space $ (X,d)$ to be compact, it is not enough that $A$ is totally bounded and closed (since $X$ is always closed). However, the correct assumptions to conclude that $A$ is compact are that it is totally bounded and complete.
Compact space - Wikipedia
WebAug 1, 2024 · No unbounded set or not closed set can be compact in any metric space. Solution 2 Boundedness Part of the problem is that boundedness is a nearly useless property by itself in the context of metric spaces. Consider a metric space ( X, d) and define a new metric b on X by b ( x, y) := min { d ( x, y), 1 }. WebWhy Closed, Bounded Sets in \n are Compact Suppose A is a closed, bounded subset of \n. Then ∃ M>0 such that A⊂{(x1,…xn)∈ \ n: x j ≤M, ∀ j}=B. That A is compact will follow … how to make java rice aristocrat style
Why Closed, Bounded Sets in n are Compact - UCLA …
WebThe interval C = (2, 4) is not compact because it is not closed (but bounded). The interval B = [0, 1] is compact because it is both closed and bounded. In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. [1] WebOct 17, 2024 · How can I prove that the interval $[0,∞)$ is closed and bounded in $(\mathbb{R},d)$ but not compact under the distance function $ d(x, y) = \min \{ x − y ,1 … WebAug 1, 2024 · No unbounded set or not closed set can be compact in any metric space. Solution 2 Boundedness Part of the problem is that boundedness is a nearly useless … how to make java server cross platform