Exploring Compactness and Connectedness

Exploring Compactness and Connectedness

University

20 Qs

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Exploring Compactness and Connectedness

Exploring Compactness and Connectedness

Assessment

Quiz

Mathematics

University

Practice Problem

Medium

Created by

Stephy Stephen

Used 4+ times

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20 questions

Show all answers

1.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Define compactness in a metric space.

A subset is compact if it is closed and bounded.

A compact metric space has a finite number of points.

A subset of a metric space is compact if every open cover has a finite subcover.

A subset is compact if it contains all its limit points.

2.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is a connected space? Provide an example.

The set of all integers is a connected space.

The interval [0, 1] in the real numbers is a connected space.

The interval (-1, 1) in the real numbers is a disconnected space.

The empty set is a connected space.

3.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Prove that the closed interval [0, 1] is compact.

The interval [0, 1] is not closed.

The interval [0, 1] does not contain its endpoints.

The closed interval [0, 1] is compact.

The interval [0, 1] is open and unbounded.

4.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Identify whether the set of all rational numbers is connected.

The set of all rational numbers is partially connected.

The set of all rational numbers is not connected.

The set of all rational numbers is connected.

The set of all rational numbers is completely connected.

5.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Explain the Heine-Borel theorem in relation to compactness.

A set is compact if it is open and bounded.

A set is compact if it is finite and bounded.

In Euclidean space, a set is compact if and only if it is closed and bounded.

A set is compact if it is closed and unbounded.

6.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Prove that a continuous image of a compact space is compact.

The image of a compact space is always empty.

A continuous image of a compact space is compact.

A continuous image of a non-compact space is compact.

A compact space cannot be continuous.

7.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the definition of a normed linear space?

A normed linear space is a collection of scalar values.

A normed linear space is a geometric shape in three-dimensional space.

A normed linear space is a type of matrix.

A normed linear space is a vector space with a norm that measures the size of vectors.

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