Concept Mastery Quiz On Induced Topology In Metric Spaces

1) If Y is a closed subset of X, then every open set in Y is the intersection of a closed set in X with Y.

True

False

False, because the subspace topology is defined by intersecting open sets of X with Y.

Explanation

False, because the subspace topology is defined by intersecting open sets of X with Y.

2) Let X = ℝ, Y = ℤ. Which sets are open in the induced topology on Y?

{1,2,3}

ℤ

{n} for any n ∈ ℤ

∅

The induced topology on ℤ is discrete, so every subset is open.

Explanation

The induced topology on ℤ is discrete, so every subset is open.

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3) The induced topology of a subspace of a compact metric space is always compact.

True

False

False, because a subspace-open set may not be open in X (e.g., [0,0.5) in Y).

Explanation

False, because a subspace-open set may not be open in X (e.g., [0,0.5) in Y).

4) Let X = ℝ, Y = (0,1). Which set is not open in the induced topology on Y?

(0.2,0.8)

(0,0.5)

[0.3,0.7]

(0.1,0.9)

[0.3,0.7] cannot be expressed as an intersection of an open set in ℝ with Y.

Explanation

[0.3,0.7] cannot be expressed as an intersection of an open set in ℝ with Y.

5) The induced topology on a finite set is always discrete.

True

False

True—every singleton arises as an intersection of an open set with the finite set.

Explanation

True—every singleton arises as an intersection of an open set with the finite set.

6) Let X = ℝ, Y = ℚ. Which statement is correct?

Every interval (a,b) ∩ ℚ is open in ℚ

ℚ is closed in its own induced topology

ℚ has the discrete topology

No singleton is open in ℚ

Subspace-open sets in ℚ are intersections with open intervals. ℚ is always closed in itself.

Explanation

Subspace-open sets in ℚ are intersections with open intervals. ℚ is always closed in itself.

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7) The closure of a set in a subspace equals the intersection of the closure of the set in the ambient space with the subspace.

True

False

True—this is a fundamental closure property of subspaces.

Explanation

True—this is a fundamental closure property of subspaces.

8) Let X = ℝ², Y = {(x,y): xy = 0}. Which sets are open in the induced topology on Y?

{(x,0): x > 0}

{(0,y): y < 0}

{(0,0)}

{(x,0): x ∈ ℝ}

These are intersections of open regions in ℝ² with Y; a single point is never open.

Explanation

These are intersections of open regions in ℝ² with Y; a single point is never open.

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9) If f:X→Z is continuous and Y ⊆ X, then the restriction f|Y: Y→Z is continuous with respect to the induced topology.

True

False

True—continuity remains preserved when restricting the domain.

Explanation

True—continuity remains preserved when restricting the domain.

10) If Y,d_y) is a subspace of a metric space (X,d_x), then the open sets in Y are exactly the intersections of open sets in X with Y.

True

False

True, because the subspace topology is defined by intersecting open sets of X with Y.

Explanation

True, because the subspace topology is defined by intersecting open sets of X with Y.

11) Let X=ℝ with the usual metric, and let Y=[0,1]⊂X. Which of the following sets is open in the induced topology on Y?

[0,0.5)

[0,0.6)

[0.5,1]

[0,1]

In the induced topology on the subset Y=[0,1], a set is open if it can be expressed as the intersection of an open set in X with Y. The set [0,0.5) is open in Y since it can be represented as the intersection of the open interval (-∞, 0.5) in X with Y, resulting in [0,0.5). In contrast, the sets [0,0.6), [0.5,1], and [0,1] either include points that are not in the interior of Y or do not meet the criteria for openness in the induced topology on Y.

Explanation

In the induced topology on the subset Y=[0,1], a set is open if it can be expressed as the intersection of an open set in X with Y. The set [0,0.5) is open in Y since it can be represented as the intersection of the open interval (-∞, 0.5) in X with Y, resulting in [0,0.5). In contrast, the sets [0,0.6), [0.5,1], and [0,1] either include points that are not in the interior of Y or do not meet the criteria for openness in the induced topology on Y.

12) Every open set in the induced topology is also open in the original metric space.

True

False

False, because a subspace-open set may not be open in X (e.g., [0,0.5) in Y).

Explanation

False, because a subspace-open set may not be open in X (e.g., [0,0.5) in Y).

13) Let (X,d) be a metric space and YX. Which of the following is always true?

Every closed set in Y is closed in X

The empty set is open in Y

Every singleton in Y is open in Y

Y itself cannot be open in the induced topology

In a metric space, closed sets in a subspace Y are defined in relation to the topology of X. Since closed sets in Y are intersections of closed sets in X with Y, it follows that every closed set in Y is also closed in X. Hence, option A is always true, while the other options may not hold in general.

Explanation

In a metric space, closed sets in a subspace Y are defined in relation to the topology of X. Since closed sets in Y are intersections of closed sets in X with Y, it follows that every closed set in Y is also closed in X. Hence, option A is always true, while the other options may not hold in general.