Problem-Solving And Reasoning Quiz On Induced Topology

1) Let X = ℝ and Y = [0,2]. Which of the following sets is open in the induced topology on Y?

(0.5,1.5)

(1,3)

(-1,1)

[1,2]

(0.5,1.5) is the intersection of an open interval with Y; others are not.

Explanation

(0.5,1.5) is the intersection of an open interval with Y; others are not.

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2) Let X = ℝ² and Y = {(x,0) : x ∈ ℝ}. Which set must be open in the induced topology on Y?

(3,5) × {0}

A single point (1,0)

ℝ × {0}

{(x,0): x>0}

These are intersections of open sets in ℝ² with Y. Singletons are not open.

Explanation

These are intersections of open sets in ℝ² with Y. Singletons are not open.

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3) If A ⊆ Y is open in Y, then:

There exists open U in X with A = U ∩ Y

A is open in X

Y\A closed in X

Y\A closed in Y

Open in Y means A = U ∩ Y; complements of open sets are closed in Y.

Explanation

Open in Y means A = U ∩ Y; complements of open sets are closed in Y.

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4) Let X = ℝ² and Y be the unit circle. Which sets are closed in the induced topology on Y?

A finite set of points

The entire circle

Any arc

Arc’s complement

Finite sets, whole circle, and arc complements are closed in Y.

Explanation

Finite sets, whole circle, and arc complements are closed in Y.

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

All singletons {n}

Any finite subset of ℤ

Any subset of ℤ

ℤ

ℤ inherits a discrete topology; all subsets are open.

Explanation

ℤ inherits a discrete topology; all subsets are open.

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6) Let X = ℝ and Y = (0,1) ∪ (2,3). Which is open in Y?

(0,1)

(2,2.5)

[0.2,0.4)

(1,2)∩Y

Only intersections of open intervals with Y are open.

Explanation

Only intersections of open intervals with Y are open.

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7) If Y ⊆ X is dense in X, which must be true?

Every open set in Y is dense in X

A set closed in Y might not be closed in X

Closure in X = closure in Y

Induced topology same as X

Closed in Y does not imply closed in X.

Explanation

Closed in Y does not imply closed in X.

8) Let Y ⊆ X. Which can occur only because of induced topology effects?

A set open in Y but not X

A set closed in X but not Y

Sequence converges in Y but diverges in X

Singleton becomes open in Y

Subspace effects allow openness in Y but not in X; singletons can become open.

Explanation

Subspace effects allow openness in Y but not in X; singletons can become open.

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9) Let X = ℝ and Y = [0,2]. Which sets are closed in Y?

[0,1.5]

(0.5,2)

{1}

(-∞,1] ∩ Y

These come from intersections with closed sets in ℝ.

Explanation

These come from intersections with closed sets in ℝ.

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10) Let X = ℝ² and Y = {(x,y): y = 2x}. Which subsets of Y are open in the induced topology?

A single point

Any segment (a<x<b)

{(x,2x): x ≥ 0}

The entire line Y

Segments and the whole line Y are open; singletons are not.

Explanation

Segments and the whole line Y are open; singletons are not.

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11) A student claims: 'Every open ball in Y is also an open ball in X.' Why is this incorrect?

Open balls in Y = intersections with X balls

Y may cut the ball into non-ball shapes

Open balls in Y may be smaller

Every open ball in X is in Y

Subspace balls are intersections and may be cut or smaller.

Explanation

Subspace balls are intersections and may be cut or smaller.

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12) Let X = ℝ and Y = [0,1] ∪ (2,3). Which is true?

{1}

(2.1,2.8)

(0.2,0.8)

Y is open in itself

Intersections with open intervals remain open; Y is open in itself.

Explanation

Intersections with open intervals remain open; Y is open in itself.

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13) Let X = ℝ² and Y be the unit circle. Which set is closed in the induced topology on Y?

A single point on the circle

A semicircle

Any finite set of points

Empty set

Finite sets and ∅ are closed; semicircles aren't.

Explanation

Finite sets and ∅ are closed; semicircles aren't.

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14) Let X = ℝ² and Y be a closed disk. If F ⊆ Y is closed in the induced topology, which must be true?

There is a closed set C⊂X such that F=C∩Y.

F contains limits of sequences in Y

F must be closed in X

Y\F is open in Y

F = C∩Y, contains its limits, and complement is open in Y.

Explanation

F = C∩Y, contains its limits, and complement is open in Y.

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15) Let Y = {(x,y): x² + y² = 1}. Student claims: 'Every connected open subset of Y is the entire circle.' Is this correct?

Yes, because circles are connected

No, because arcs are open in the subspace topology and are connected.

Yes, because open subsets must contain interior points

No, because open neighborhoods in Y are 1-dimensional.

Arcs are connected open sets in Y; circle topology is 1D.

Explanation

Arcs are connected open sets in Y; circle topology is 1D.

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Problem-Solving and Reasoning Quiz on Induced Topology

1) Let X = ℝ and Y = [0,2]. Which of the following sets is open in the induced topology on Y?

2)

What first name or nickname would you like us to use?

2) Let X = ℝ² and Y = {(x,0) : x ∈ ℝ}. Which set must be open in the induced topology on Y?

3) If A ⊆ Y is open in Y, then:

4) Let X = ℝ² and Y be the unit circle. Which sets are closed in the induced topology on Y?

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

6) Let X = ℝ and Y = (0,1) ∪ (2,3). Which is open in Y?

7) If Y ⊆ X is dense in X, which must be true?

8) Let Y ⊆ X. Which can occur only because of induced topology effects?

9) Let X = ℝ and Y = [0,2]. Which sets are closed in Y?

10) Let X = ℝ² and Y = {(x,y): y = 2x}. Which subsets of Y are open in the induced topology?

11) A student claims: 'Every open ball in Y is also an open ball in X.' Why is this incorrect?

12) Let X = ℝ and Y = [0,1] ∪ (2,3). Which is true?

13) Let X = ℝ² and Y be the unit circle. Which set is closed in the induced topology on Y?

14) Let X = ℝ² and Y be a closed disk. If F ⊆ Y is closed in the induced topology, which must be true?

15) Let Y = {(x,y): x² + y² = 1}. Student claims: 'Every connected open subset of Y is the entire circle.' Is this correct?