# Example of a Quotient Map That Is Neither Open Nor Closed

• jmjlt88
In summary, the conversation discusses an example of a quotient map that is not open nor closed. It introduces a projection map and a subspace A consisting of certain coordinates. It is proven that q, a restriction of π, is not a closed or open map. The example of [0,∞)x (-∞,0) is used to illustrate this. The conversation also asks for clarification on whether the example is correct or not.
jmjlt88
We are just looking for an example of a quotient map that is not open nor closed. Let π: ℝxℝ -> ℝ be a projection onto the first coordinate. Let A be the subspace of ℝxℝ consisting of all points (x,y) such that x≥0 or y=0 or both. Let q:A -> ℝ be a restriction of π. ( Note: assume that q was already proved to be, in fact, a quotient map.) The set ℝx{0} is a subset of A and it is closed in ℝxℝ. Since ℝx{0} is the intersection of A with a closed set of ℝxℝ, it is closed in A. Then, q (ℝx{0}) = ℝ, which is open. Hence, q is not a closed map. Since [0,∞)x (-∞,0) is a subset of A, and it is the intersection of A with ℝx(-∞,0), an open set in ℝxℝ, [0,∞)x (-∞,0) is open in A. However, q ([0,∞)x (-∞,0)) = [0,∞), which is closed. Hence, q is not an open map.

Is this close?

Thank you! :shy:

jmjlt88 said:
We are just looking for an example of a quotient map that is not open nor closed. Let π: ℝxℝ -> ℝ be a projection onto the first coordinate. Let A be the subspace of ℝxℝ consisting of all points (x,y) such that x≥0 or y=0 or both. Let q:A -> ℝ be a restriction of π. ( Note: assume that q was already proved to be, in fact, a quotient map.) The set ℝx{0} is a subset of A and it is closed in ℝxℝ. Since ℝx{0} is the intersection of A with a closed set of ℝxℝ, it is closed in A. Then, q (ℝx{0}) = ℝ, which is open. Hence, q is not a closed map. Since [0,∞)x (-∞,0) is a subset of A, and it is the intersection of A with ℝx(-∞,0), an open set in ℝxℝ, [0,∞)x (-∞,0) is open in A. However, q ([0,∞)x (-∞,0)) = [0,∞), which is closed. Hence, q is not an open map.

Is this close?

Thank you! :shy:

R is both open and closed in R, so the first part doesn't work. Think about the graph G of y=1/x for x>0. Is G open or closed? What about q(G)?

## 1. What is a quotient map?

A quotient map is a type of function in mathematics that maps points from one space onto another while preserving certain properties, such as topological structure or algebraic properties.

## 2. What does it mean for a quotient map to be open?

A quotient map is open if it maps open sets to open sets. This means that the preimage of an open set in the target space is an open set in the original space.

## 3. What does it mean for a quotient map to be closed?

A quotient map is closed if it maps closed sets to closed sets. This means that the preimage of a closed set in the target space is a closed set in the original space.

## 4. Can a quotient map be both open and closed?

Yes, it is possible for a quotient map to be both open and closed. This would mean that it maps both open and closed sets to open and closed sets, respectively.

## 5. Can you give an example of a quotient map that is neither open nor closed?

Yes, an example of a quotient map that is neither open nor closed is the projection map from the real line to the circle. This map is not open because it maps open intervals in the real line to arcs on the circle, which are not open sets. It is also not closed because it maps closed intervals in the real line to arcs on the circle, which are not closed sets.

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