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jmjlt88

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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.

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jmjlt88

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Is this close?

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Dick

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jmjlt88 said:

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)?

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.

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.

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.

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.

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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