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Topological Spaces and Continuous Functions

Description: This quiz is designed to assess your understanding of the fundamental concepts related to Topological Spaces and Continuous Functions. The questions cover topics such as open and closed sets, continuity, compactness, and connectedness.
Number of Questions: 14
Created by:
Tags: topology continuous functions open sets closed sets compactness connectedness
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Which of the following is NOT a property of open sets in a topological space?

  1. They contain all their limit points.

  2. They are closed under arbitrary unions.

  3. They are closed under finite intersections.

  4. They contain no isolated points.

Show answer
Correct Option: A
Explanation:

Open sets in a topological space do not necessarily contain all their limit points. For example, in the real line with the usual topology, the open interval (0, 1) does not contain its limit point 1.

Let $f: X \rightarrow Y$ be a function between two topological spaces. Which of the following statements is true?

  1. $f$ is continuous if and only if the preimage of every open set in $Y$ is an open set in $X$.

  2. $f$ is continuous if and only if the preimage of every closed set in $Y$ is a closed set in $X$.

  3. $f$ is continuous if and only if the image of every open set in $X$ is an open set in $Y$.

  4. $f$ is continuous if and only if the image of every closed set in $X$ is a closed set in $Y$.

Show answer
Correct Option: A
Explanation:

A function $f: X \rightarrow Y$ between two topological spaces is continuous if and only if the preimage of every open set in $Y$ is an open set in $X$.

Which of the following is NOT a property of compact topological spaces?

  1. Every open cover has a finite subcover.

  2. Every infinite subset has a limit point.

  3. Every continuous function from a compact space to a Hausdorff space is uniformly continuous.

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

Show answer
Correct Option: B
Explanation:

Compact topological spaces do not necessarily have the property that every infinite subset has a limit point. For example, the Cantor set is a compact topological space that does not contain any isolated points, and therefore, every infinite subset of the Cantor set has no limit point.

Which of the following statements is true about connected topological spaces?

  1. Every connected space is path-connected.

  2. Every path-connected space is connected.

  3. Every connected space is locally connected.

  4. Every locally connected space is connected.

Show answer
Correct Option: B
Explanation:

Every path-connected space is connected, but the converse is not necessarily true. For example, the space consisting of two disjoint open intervals is connected but not path-connected.

Let $X$ be a topological space and $A \subseteq X$. Which of the following statements is true about the closure of $A$?

  1. It is the smallest closed set containing $A$.

  2. It is the largest open set containing $A$.

  3. It is the union of $A$ and all its limit points.

  4. It is the intersection of $A$ and all its limit points.

Show answer
Correct Option: A
Explanation:

The closure of a set $A$ in a topological space $X$ is the smallest closed set containing $A$.

Let $X$ be a topological space and $f: X \rightarrow Y$ be a continuous function. Which of the following statements is true?

  1. The image of a connected set under $f$ is connected.

  2. The image of a compact set under $f$ is compact.

  3. The preimage of a connected set under $f$ is connected.

  4. The preimage of a compact set under $f$ is compact.

Show answer
Correct Option: A
Explanation:

The image of a connected set under a continuous function is connected.

Which of the following is an example of a topological space that is not Hausdorff?

  1. The real line with the usual topology

  2. The Euclidean plane with the usual topology

  3. The set of rational numbers with the usual topology

  4. The Sorgenfrey line

Show answer
Correct Option: D
Explanation:

The Sorgenfrey line is an example of a topological space that is not Hausdorff. It is a topological space where every point is a limit point of every non-empty open set.

Let $X$ be a topological space and $A \subseteq X$. Which of the following statements is true about the interior of $A$?

  1. It is the largest open set contained in $A$.

  2. It is the smallest closed set containing $A$.

  3. It is the union of $A$ and all its limit points.

  4. It is the intersection of $A$ and all its limit points.

Show answer
Correct Option: A
Explanation:

The interior of a set $A$ in a topological space $X$ is the largest open set contained in $A$.

Which of the following is an example of a topological space that is not compact?

  1. The closed interval [0, 1] with the usual topology

  2. The open interval (0, 1) with the usual topology

  3. The set of integers with the usual topology

  4. The Cantor set

Show answer
Correct Option: B
Explanation:

The open interval (0, 1) with the usual topology is an example of a topological space that is not compact. It is not compact because it does not have the property that every open cover has a finite subcover.

Let $X$ be a topological space and $f: X \rightarrow Y$ be a continuous function. Which of the following statements is true?

  1. The preimage of a closed set under $f$ is closed.

  2. The preimage of an open set under $f$ is open.

  3. The image of a closed set under $f$ is closed.

  4. The image of an open set under $f$ is open.

Show answer
Correct Option: A
Explanation:

The preimage of a closed set under a continuous function is closed.

Which of the following is an example of a topological space that is both compact and connected?

  1. The closed interval [0, 1] with the usual topology

  2. The open interval (0, 1) with the usual topology

  3. The set of integers with the usual topology

  4. The Cantor set

Show answer
Correct Option: A
Explanation:

The closed interval [0, 1] with the usual topology is an example of a topological space that is both compact and connected.

Let $X$ be a topological space and $A \subseteq X$. Which of the following statements is true about the boundary of $A$?

  1. It is the set of points that are in the closure of $A$ but not in the interior of $A$.

  2. It is the set of points that are in the interior of $A$ but not in the closure of $A$.

  3. It is the set of points that are in both the closure of $A$ and the interior of $A$.

  4. It is the set of points that are in neither the closure of $A$ nor the interior of $A$.

Show answer
Correct Option: A
Explanation:

The boundary of a set $A$ in a topological space $X$ is the set of points that are in the closure of $A$ but not in the interior of $A$.

Which of the following is an example of a topological space that is not locally compact?

  1. The real line with the usual topology

  2. The Euclidean plane with the usual topology

  3. The set of rational numbers with the usual topology

  4. The Sorgenfrey line

Show answer
Correct Option: D
Explanation:

The Sorgenfrey line is an example of a topological space that is not locally compact. It is not locally compact because there exists a point $x$ in the Sorgenfrey line such that every neighborhood of $x$ contains a non-compact set.

Let $X$ be a topological space and $f: X \rightarrow Y$ be a continuous function. Which of the following statements is true?

  1. If $f$ is one-to-one, then $f$ is a homeomorphism.

  2. If $f$ is onto, then $f$ is a homeomorphism.

  3. If $f$ is both one-to-one and onto, then $f$ is a homeomorphism.

  4. None of the above.

Show answer
Correct Option: C
Explanation:

A function $f: X \rightarrow Y$ between two topological spaces is a homeomorphism if and only if $f$ is both one-to-one and onto.

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