Let (X, d) be a metric sparse and let B be a subset of X then if B is closed then B is also ______.
Understanding Metric Spaces, Closed Sets, and Complete Sets
The question asks about a property of a closed subset within a metric space. It states "Let (X, d) be a metric sparse space and let B be a subset of X then if B is closed then B is also ______." The term "metric sparse space" is not standard terminology in topology or analysis. It is highly probable that "sparse" is a typographical error and the question intended to refer to a "complete metric space". We will proceed by analyzing the question under the assumption that X is a complete metric space, as the provided answer aligns with a well-known theorem in this context.
What is a Metric Space?
A metric space is a set X equipped with a function d (called a metric or distance function) that satisfies the following properties for all x, y, z in X:
- Non-negativity: $d(x, y) \ge 0$
- Identity of indiscernibles: $d(x, y) = 0$ if and only if $x = y$
- Symmetry: $d(x, y) = d(y, x)$
- Triangle inequality: $d(x, z) \le d(x, y) + d(y, z)$
What is a Closed Subset?
A subset B of a metric space (X, d) is called closed if it contains all its limit points. Equivalently, B is closed if its complement X \ B is an open set in X.
What is a Complete Metric Space?
A metric space (X, d) is called complete if every Cauchy sequence in X converges to a point in X.
- A sequence $(x_n)$ in X is a Cauchy sequence if for every $\epsilon > 0$, there exists an integer N such that for all m, n > N, $d(x_m, x_n) < \epsilon$.
- A sequence $(x_n)$ in X converges to a point x $\in$ X if for every $\epsilon > 0$, there exists an integer N such that for all n > N, $d(x_n, x) < \epsilon$.
Analyzing the Relationship Between Closed Sets and Complete Sets
The crucial theorem related to this question (assuming X is complete) is:
Theorem: Every closed subset of a complete metric space is complete.
Let's understand why this theorem is true. Suppose (X, d) is a complete metric space, and B is a closed subset of X. Let $(b_n)$ be a Cauchy sequence in B. Since B is a subset of X, $(b_n)$ is also a Cauchy sequence in X. Because X is complete, the sequence $(b_n)$ converges to some point, say $x$, in X. Since B is closed and $(b_n)$ is a sequence in B converging to $x$, the limit point $x$ must belong to B. Therefore, every Cauchy sequence in B converges to a point in B, which means B is complete with the induced metric from X.
Evaluating the Options
Given the likely interpretation that (X, d) is a complete metric space and B is a closed subset:
- Unbounded: A closed set in a complete metric space is not necessarily unbounded. For example, the closed interval [0, 1] is closed and bounded in the complete metric space ℝ.
- Complete: As explained by the theorem, a closed subset of a complete metric space is complete. This aligns with the provided correct answer.
- Dense: A closed set is not necessarily dense. A set B is dense in X if its closure $\bar{B}$ is equal to X. If B is closed, $\bar{B} = B$. So B is dense if and only if B = X. A proper closed subset (like {0} in ℝ) is closed but not dense.
- Bounded: A closed set in a complete metric space is not necessarily bounded. For example, the set of integers $\mathbb{Z}$ is closed in the complete metric space ℝ but it is not bounded.
Based on the analysis and the standard theorem, if we assume (X, d) is a complete metric space, then a closed subset B of X is also complete.
Thus, if B is closed and (X, d) is a complete metric space, then B is also complete.
| Property of Subset B | Does it imply B is Complete (if X is complete)? | Example (in ℝ) |
|---|---|---|
| Closed | Yes (Theorem: A closed subset of a complete metric space is complete) | [0, 1] is closed in ℝ (complete), and [0, 1] is complete. |
| Open | No | (0, 1) is open in ℝ (complete), but (0, 1) is not complete. |
| Bounded | No (unless also closed and X=R$^n$ or similar space) | (0, 1) is bounded in ℝ, but (0, 1) is not complete. |
Revision Table: Metric Space Concepts
| Term | Definition | Relevance to the Question |
|---|---|---|
| Metric Space (X, d) | A set X with a distance function d satisfying specific properties. | The space where the subset B exists. The property of X (like completeness) affects properties of its subsets. |
| Closed Set B $\subseteq$ X | Contains all its limit points; its complement is open. | This is the given property of the subset B in the question. |
| Complete Metric Space | Every Cauchy sequence in the space converges to a point within the space. | If the parent space X is complete, it imparts the property of completeness to its closed subsets. |
| Cauchy Sequence | A sequence where terms become arbitrarily close to each other as the sequence progresses. | Used in the definition of completeness. |
| Convergent Sequence | A sequence that approaches a specific limit point. | Completeness requires Cauchy sequences to be convergent within the space. |
Additional Information on Completeness
Completeness is a very important property in metric spaces. It guarantees that there are "no holes" in the space in terms of convergence of sequences. Here are some related facts:
- The space of real numbers ℝ with the standard metric $d(x,y) = |x-y|$ is complete.
- The space of rational numbers ℚ with the standard metric is not complete. For example, a sequence of rational numbers converging to $\sqrt{2}$ (which is irrational) is a Cauchy sequence in ℚ but does not converge to a point in ℚ.
- A subset B of a metric space (X, d) is complete if and only if it is closed in X and the metric space (B, d$_B$) is complete, where d$_B$ is the restricted metric.
- The theorem used in the solution, "A closed subset of a complete metric space is complete," is a fundamental result. However, the converse is not true: a complete subset of a metric space is always closed, regardless of whether the parent space X is complete.
- Compact metric spaces are always complete.
Assuming "sparse" was a typo for "complete", the property of B being closed implies it inherits the completeness from the parent space X.
Important Questions from Metric Spaces
Let (X, d) be a compact metric space. Let T ∶ X → X be a continuous function satisfying infn∈ℕ d(Tn(x),Tn (y)) ≠ 0 for every x, y ∈ X with x ≠ y. Then which of the following statements are true?
For any two metric spaces (X, dX), (Y, dY) a map f ∶ X → Y is said to be a closed map if whenever F is closed in X, then f(F) is closed in Y. For any subset B of a metric space, B is given the induced metric. The metric on X × Y is given by d((x, y), (x', y')) = max{dX(x, x'), dY(y, y')}. Which of the following are true?
Let p ∶ \(\mathbb{R}\)2 → \(\mathbb{R}\) be defined by
\(p(x, y)=\left\{\begin{array}{lll} |x| & \text { if } & x \neq 0 \\ |y| & \text { if } & x=0 . \end{array}\right.\)
Which of the following statements are true?
Let X = \(\rm\displaystyle\prod_{n=1}^{\infty}\)[0, 1], that is, the space of sequences {xn}n ≥ 1 with xn ∈ [0, 1], n ≥ 1.
Define the metric d: X × X → [0, ∞) by
d({xn}n≥1, {yn}n≥1) = \(\rm\displaystyle\sup _{n ≥ 1} \frac{\left|x_n-y_n\right|}{2^n}\).
Which of the following statements are true?
If f is continuous mapping of compact metric space X into a metric space Y then:
