Exploring Compactness and Connectedness

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.

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.

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.

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.

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.

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.

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.

Give an example of a normed linear space and its norm.

• L^2 space with the norm ||f|| = ∫|f(x)|^2 dx

• C[0, 1] with the norm ||f|| = max{|f(x)| : x ∈ [0, 1}

• C[0, 1] with the norm ||f|| = ∫|f(x)| dx

• R^2 with the norm ||v|| = |v_1| + |v_2|

Prove that every finite-dimensional normed linear space is complete.

• Every finite-dimensional normed linear space is complete.

• Some finite-dimensional normed linear spaces are incomplete.

• Completeness is not a property of normed linear spaces.

• Every infinite-dimensional normed linear space is complete.

What is the relationship between compactness and continuity?

• Compactness ensures that continuous functions behave well, particularly in achieving extrema.

• Compactness only applies to discrete functions.

• Continuous functions cannot achieve extrema on compact sets.

• Compactness is irrelevant to continuity.