Definition Mastery Quiz On Subspace And Induced Topology

1) If Y⊆X, the induced topology on Y always contains fewer open sets than the topology of X.

True

False

False, because Y may have more open sets relative to itself than X does (e.g., Y=ℤ becomes discrete).

Explanation

False, because Y may have more open sets relative to itself than X does (e.g., Y=ℤ becomes discrete).

2) A set that is open in a metric subspace Y must be the intersection of some open ball in X with Y.

True

False

True, because subspace-open sets are intersections of open sets in X with Y.

Explanation

True, because subspace-open sets are intersections of open sets in X with Y.

3) If Y⊆X and A⊆Y, then A is closed in Y iff there exists a closed set C⊆X such that A=C∩Y.

True

False

True, this is the definition of closed sets in the induced topology.

Explanation

True, this is the definition of closed sets in the induced topology.

4) If Y⊆X is open, then every open set in the induced topology on Y is also open in X.

True

False

True, because intersections of open sets with Y remain open in X when Y is open.

Explanation

True, because intersections of open sets with Y remain open in X when Y is open.

5) The induced topology on any infinite subset Y⊆ℝ is always non-discrete.

True

False

False, an infinite set like ℤ inherits the discrete topology.

Explanation

False, an infinite set like ℤ inherits the discrete topology.

6) If Y⊆X is dense in X, then the induced topology on Y must be the same as the topology on X.

True

False

False, Y does not inherit all opens of X; e.g., ℚ is dense in ℝ but not topologically identical.

Explanation

False, Y does not inherit all opens of X; e.g., ℚ is dense in ℝ but not topologically identical.

7) If (X,d) is complete and Y⊆X, then Y with the induced metric is always complete.

True

False

False, only closed subsets of complete spaces are complete.

Explanation

False, only closed subsets of complete spaces are complete.

8) Let X=ℝ, Y=[2,5]. Which must be open in the induced topology on Y?

[2,3)

(1,4)

[3,4]

(2.5,4.5)∩Y

These are intersections of open intervals in ℝ with Y.

Explanation

These are intersections of open intervals in ℝ with Y.

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9) If X=ℝ, Y=(0,1). Which is open in Y?

(0,0.3)

[0.1,0.2]

[0,0.4)

(0.5,1)

Only intersections of open intervals with Y are open.

Explanation

Only intersections of open intervals with Y are open.

10) Let X=ℝ², Y={(x,0):x∈ℝ}. Which sets are open in Y?

{(x,0):x>2}

{(x,0):x=0}

{(0,0)}

Y

A is an open interval on the line; Y is open in itself.

Explanation

A is an open interval on the line; Y is open in itself.

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11) Let X=ℝ, Y={1/n : n∈ℕ}. Which is open in Y?

{1/2}

{1/100,1/200}

Y{1}

∅

Y inherits the discrete topology; all subsets are open.

Explanation

Y inherits the discrete topology; all subsets are open.

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12) Which statement is always true for the induced topology on Y⊆X?

Finite intersections open

Every subset closed

Open sets restrict

Open in Y => open in X

Finite intersections remain open, and restrictions of open sets stay open.

Explanation

Finite intersections remain open, and restrictions of open sets stay open.

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13) Let X=ℝ², Y={(x,y):y=|x|}. Which sets are open in Y?

{(x,|x|):x>0}

{(0,0)}

Y

{(x,|x|):|x|<1}

These come from intersecting open planar sets with Y; singletons are not open.

Explanation

These come from intersecting open planar sets with Y; singletons are not open.

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14) Let X=ℝ², Y=S¹. Which is NOT open in the induced topology?

A semicircle

Arc

Single point

Whole circle

Single points are not open in the 1-dimensional circle topology.

Explanation

Single points are not open in the 1-dimensional circle topology.

15) Suppose X=ℝ and Y=ℝ\ℚ. Which is open in Y?

(a,b)∩Y

Y

(a,b)

Singleton

Open sets in Y must be intersections with open sets in ℝ; singletons are not open.

Explanation

Open sets in Y must be intersections with open sets in ℝ; singletons are not open.

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Definition Mastery Quiz on Subspace and Induced Topology

1) If Y⊆X, the induced topology on Y always contains fewer open sets than the topology of X.

2)

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

2) A set that is open in a metric subspace Y must be the intersection of some open ball in X with Y.

3) If Y⊆X and A⊆Y, then A is closed in Y iff there exists a closed set C⊆X such that A=C∩Y.

4) If Y⊆X is open, then every open set in the induced topology on Y is also open in X.

5) The induced topology on any infinite subset Y⊆ℝ is always non-discrete.

6) If Y⊆X is dense in X, then the induced topology on Y must be the same as the topology on X.

7) If (X,d) is complete and Y⊆X, then Y with the induced metric is always complete.

8) Let X=ℝ, Y=[2,5]. Which must be open in the induced topology on Y?

9) If X=ℝ, Y=(0,1). Which is open in Y?

10) Let X=ℝ², Y={(x,0):x∈ℝ}. Which sets are open in Y?

11) Let X=ℝ, Y={1/n : n∈ℕ}. Which is open in Y?

12) Which statement is always true for the induced topology on Y⊆X?

13) Let X=ℝ², Y={(x,y):y=|x|}. Which sets are open in Y?

14) Let X=ℝ², Y=S¹. Which is NOT open in the induced topology?

15) Suppose X=ℝ and Y=ℝ\ℚ. Which is open in Y?