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Topological Spaces

Description: This quiz covers the fundamental concepts and properties of topological spaces, including open and closed sets, continuity, and compactness.
Number of Questions: 16
Created by:
Tags: topology topological spaces open sets closed sets continuity compactness
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In a topological space, a set is open if it satisfies which of the following conditions?

  1. It contains all of its limit points.

  2. It is the complement of a closed set.

  3. It is the union of open intervals.

  4. It is the intersection of open sets.

Show answer
Correct Option: B
Explanation:

In a topological space, a set is open if it is the complement of a closed set. This means that a set is open if it does not contain any of its limit points.

Which of the following statements is true about closed sets in a topological space?

  1. A set is closed if it contains all of its limit points.

  2. A set is closed if it is the complement of an open set.

  3. A set is closed if it is the union of closed sets.

  4. A set is closed if it is the intersection of closed sets.

Show answer
Correct Option: A
Explanation:

In a topological space, a set is closed if it contains all of its limit points. This means that a set is closed if every point in the boundary of the set is also in the set.

What is the definition of a continuous function between two topological spaces?

  1. A function is continuous if the preimage of every open set is open.

  2. A function is continuous if the preimage of every closed set is closed.

  3. A function is continuous if the image of every open set is open.

  4. A function is continuous if the image of every closed set is closed.

Show answer
Correct Option: A
Explanation:

A function between two topological spaces is continuous if the preimage of every open set in the codomain is open in the domain. This means that a function is continuous if it preserves the topological structure of the spaces.

Which of the following properties is equivalent to compactness in a topological space?

  1. Every open cover has a finite subcover.

  2. Every infinite subset has a limit point.

  3. Every continuous function from a compact space is bounded.

  4. Every continuous function from a compact space is uniformly continuous.

Show answer
Correct Option: A
Explanation:

In a topological space, compactness is equivalent to the property that every open cover has a finite subcover. This means that a space is compact if every collection of open sets that covers the space can be reduced to a finite collection that still covers the space.

What is the Hausdorff separation axiom in a topological space?

  1. For any two distinct points, there exist disjoint open sets containing each point.

  2. For any two distinct points, there exist open sets containing each point such that the intersection of the sets is empty.

  3. For any two distinct points, there exist open sets containing each point such that the closure of one set is disjoint from the other set.

  4. For any two distinct points, there exist open sets containing each point such that the boundary of one set is disjoint from the other set.

Show answer
Correct Option: A
Explanation:

The Hausdorff separation axiom in a topological space states that for any two distinct points, there exist disjoint open sets containing each point. This means that a space is Hausdorff if every pair of distinct points can be separated by open sets.

Which of the following spaces is not Hausdorff?

  1. The real line with the usual topology.

  2. The set of rational numbers with the usual topology.

  3. The set of integers with the discrete topology.

  4. The Cantor set with the usual topology.

Show answer
Correct Option: B
Explanation:

The set of rational numbers with the usual topology is not Hausdorff because there exist two distinct points, such as 0 and 1, that cannot be separated by disjoint open sets. This is because every open set containing 0 also contains infinitely many other rational numbers, including 1.

What is the definition of a connected topological space?

  1. A space is connected if it cannot be expressed as the union of two disjoint nonempty open sets.

  2. A space is connected if it cannot be expressed as the union of two disjoint nonempty closed sets.

  3. A space is connected if it cannot be expressed as the union of two disjoint nonempty sets.

  4. A space is connected if it cannot be expressed as the union of two disjoint nonempty subsets.

Show answer
Correct Option: A
Explanation:

A topological space is connected if it cannot be expressed as the union of two disjoint nonempty open sets. This means that a space is connected if it is not possible to divide it into two separate pieces that are both open.

Which of the following spaces is not connected?

  1. The real line with the usual topology.

  2. The set of rational numbers with the usual topology.

  3. The set of integers with the discrete topology.

  4. The Cantor set with the usual topology.

Show answer
Correct Option: B
Explanation:

The set of rational numbers with the usual topology is not connected because it can be expressed as the union of two disjoint nonempty open sets: the set of all rational numbers less than 0 and the set of all rational numbers greater than 0.

What is the definition of a compact topological space?

  1. A space is compact if every open cover has a finite subcover.

  2. A space is compact if every infinite subset has a limit point.

  3. A space is compact if every continuous function from a compact space is bounded.

  4. A space is compact if every continuous function from a compact space is uniformly continuous.

Show answer
Correct Option: A
Explanation:

A topological space is compact if every open cover of the space has a finite subcover. This means that a space is compact if it is possible to cover the space with a finite number of open sets.

Which of the following spaces is not compact?

  1. The real line with the usual topology.

  2. The set of rational numbers with the usual topology.

  3. The set of integers with the discrete topology.

  4. The Cantor set with the usual topology.

Show answer
Correct Option: A
Explanation:

The real line with the usual topology is not compact because it is not possible to cover the space with a finite number of open intervals. This is because the real line is unbounded, so there will always be some point that is not covered by any finite number of open intervals.

What is the definition of a metric space?

  1. A set with a distance function that satisfies certain properties.

  2. A set with a topology that is generated by a metric.

  3. A set with a collection of open sets that satisfy certain properties.

  4. A set with a collection of closed sets that satisfy certain properties.

Show answer
Correct Option: A
Explanation:

A metric space is a set with a distance function that satisfies certain properties, such as non-negativity, symmetry, and the triangle inequality. The distance function induces a topology on the set, which is called the metric topology.

Which of the following is not a metric space?

  1. The real line with the usual metric.

  2. The set of rational numbers with the usual metric.

  3. The set of integers with the discrete metric.

  4. The Cantor set with the usual metric.

Show answer
Correct Option: C
Explanation:

The set of integers with the discrete metric is not a metric space because the distance between any two distinct points is 1, which violates the triangle inequality.

What is the definition of a continuous function between two metric spaces?

  1. A function is continuous if the preimage of every open set is open.

  2. A function is continuous if the preimage of every closed set is closed.

  3. A function is continuous if the image of every open set is open.

  4. A function is continuous if the image of every closed set is closed.

Show answer
Correct Option: A
Explanation:

A function between two metric spaces is continuous if the preimage of every open set in the codomain is open in the domain. This means that a function is continuous if it preserves the topological structure of the spaces.

Which of the following functions is not continuous?

  1. The function f(x) = x^2 from the real line to the real line.

  2. The function f(x) = 1/x from the real line to the real line.

  3. The function f(x) = sin(x) from the real line to the real line.

  4. The function f(x) = |x| from the real line to the real line.

Show answer
Correct Option: B
Explanation:

The function f(x) = 1/x from the real line to the real line is not continuous at x = 0 because the preimage of the open set (0, 1) is not open in the domain. This is because the preimage of (0, 1) is the set of all numbers except for 0, which is not open in the real line.

What is the definition of a homeomorphism between two topological spaces?

  1. A bijective function that is continuous in both directions.

  2. A bijective function that is continuous in one direction.

  3. A function that is continuous in both directions.

  4. A function that is continuous in one direction.

Show answer
Correct Option: A
Explanation:

A homeomorphism between two topological spaces is a bijective function that is continuous in both directions. This means that a homeomorphism is a function that preserves the topological structure of the spaces.

Which of the following is not a homeomorphism?

  1. The function f(x) = x^2 from the real line to the real line.

  2. The function f(x) = 1/x from the real line to the real line.

  3. The function f(x) = sin(x) from the real line to the real line.

  4. The function f(x) = |x| from the real line to the real line.

Show answer
Correct Option: B
Explanation:

The function f(x) = 1/x from the real line to the real line is not a homeomorphism because it is not continuous at x = 0. This means that the function does not preserve the topological structure of the spaces.

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