Let (X, d) be a metric space and E is contained in X, E is called nowhere dense set if ______.
closure of E is empty
Understanding Nowhere Dense Sets in Metric Spaces
A metric space is a set where a distance function (metric) is defined, allowing us to talk about concepts like convergence, open sets, closed sets, and closure. A subset E within a metric space (X, d) can have various properties related to its density and size from a topological perspective.
The question asks for the definition of a nowhere dense set E in a metric space (X, d) based on the provided options.
Defining a Nowhere Dense Set
Based on the provided options and the identified answer text, a set E is called a nowhere dense set if:
- closure of E is empty
Let's analyze why this option is presented as the definition here, and why other options are not suitable in this context.
Analysis of Provided Options
Let's look at each option in relation to the definition of a nowhere dense set:
- closure of E is empty: According to the context of this question and the provided correct answer, this is the property that defines a nowhere dense set E. In standard mathematical definitions, the closure of a non-empty set is never empty. Only the closure of the empty set ($\overline{\emptyset}$) is empty. However, for the purpose of answering this specific question based on the provided options, this statement is considered the definition.
- closure of E terminate after some steps: This phrase does not correspond to any standard mathematical property or definition related to sets in a metric space. The closure operation is a single, well-defined process, not one that terminates in steps.
- closure of E is not defined: The closure ($\overline{E}$) of any subset E in a metric space (or any topological space) is always defined. It consists of all points in E plus all limit points of E.
- closure of E is finite: The closure of a set can be finite or infinite. For example, the closure of the set of integers {..., -1, 0, 1, ...} in the real numbers R with the usual metric is the set of integers itself, which is infinite. Also, a set can have a finite closure but not be nowhere dense (consider a single isolated point in R; its closure is just the point itself, which is finite, but its interior is the point, which is not empty). Thus, having a finite closure does not define a nowhere dense set.
Based on the analysis of the given options and the implied definition from the correct answer text, the characteristic defining a nowhere dense set in this context is that its closure is empty.
| Concept | Symbol/Notation | Description (based on context) |
|---|---|---|
| Metric Space | (X, d) | A set X with a distance function d. |
| Subset | E $\subseteq$ X | A collection of elements from X. |
| Closure of E | $\overline{E}$ | The set E combined with all its limit points. |
| Nowhere Dense Set | E | A set E where its closure is empty (according to the provided definition). |
Therefore, according to the presented options, a set E is called a nowhere dense set if the closure of E is empty.
Revision Table: Key Concepts
| Term | Brief Explanation |
|---|---|
| Metric Space | A set with a distance function. |
| Closure of a Set | The set plus its limit points. |
| Nowhere Dense Set | A set whose closure is empty (as per the specific options). |
Additional Information: Standard Mathematical Definition of Nowhere Dense Sets
It is important to note the standard mathematical definition of a nowhere dense set in topology and metric spaces. A set E in a topological space X is called nowhere dense (or meager in itself) if the interior of its closure is empty. Mathematically, this is written as $\text{int}(\overline{E}) = \emptyset$.
Let's break down the standard definition:
- First, find the closure of E ($\overline{E}$). This includes E and all its limit points.
- Then, find the interior of $\overline{E}$ ($\text{int}(\overline{E})$). The interior of a set is the largest open set contained within it.
- If this interior is the empty set ($\emptyset$), then E is nowhere dense.
Examples of nowhere dense sets under the standard definition include:
- Any finite set in R with the usual metric.
- The set of integers Z in R.
- The Cantor set in R.
These sets have non-empty closures (e.g., $\overline{\mathbb{Z}} = \mathbb{Z}$, $\overline{\text{Cantor set}} = \text{Cantor set}$), but the interior of their closures is empty (e.g., $\text{int}(\mathbb{Z}) = \emptyset$, $\text{int}(\text{Cantor set}) = \emptyset$).
The standard definition, $\text{int}(\overline{E}) = \emptyset$, is widely used in topology and functional analysis, particularly in theorems like the Baire Category Theorem, which deals with sets that are "small" or "large" from a topological viewpoint. While the provided option defines a nowhere dense set based on an empty closure, which typically only holds for the empty set itself under the standard definition, for the purpose of this specific question, we follow the provided definition.
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:
