- #1

- 1,090

- 6

First I am in [itex]\mathbb{R}[/itex] with the standard metric [d(x,y)=|y-x|]. Is [itex][0,\infty)[/itex] considered a closed set? I would think yes, since I would consider [itex](-\infty,0)[/itex] to be an open set. However, I can't seem to find any examples like this in our book, and I have yet to be able to find anything online either to clarify this. I guess I am not sure how to deal with infinity. Thoughts? Thanks!

edit... Last time I posted part of a question people wanted to see the whole thing.

So here is the question: Let [itex](\mathbb{R},d)[/itex] be the real line with the standard metric. Give an example of a continuous function [itex]f:\mathbb{R}\to\mathbb{R}[/itex], and a closed set [itex]F\subseteq \mathbb{R}[/itex], such that [itex]f(F) = \{f(x) : x \in F\}[/itex] is not closed.

So I was thinking of taking [itex]f(x)=e^x[/itex] and taking [itex](-\infty,0][/itex] as my closed set. Since that would be mapped into [itex](0,1][/itex] which is not closed. Here arises my question of is [itex](-\infty,0][/itex] closed.

Also, if you have any insightful examples for this question, I would love to see them. Thanks!

edit... Last time I posted part of a question people wanted to see the whole thing.

So here is the question: Let [itex](\mathbb{R},d)[/itex] be the real line with the standard metric. Give an example of a continuous function [itex]f:\mathbb{R}\to\mathbb{R}[/itex], and a closed set [itex]F\subseteq \mathbb{R}[/itex], such that [itex]f(F) = \{f(x) : x \in F\}[/itex] is not closed.

So I was thinking of taking [itex]f(x)=e^x[/itex] and taking [itex](-\infty,0][/itex] as my closed set. Since that would be mapped into [itex](0,1][/itex] which is not closed. Here arises my question of is [itex](-\infty,0][/itex] closed.

Also, if you have any insightful examples for this question, I would love to see them. Thanks!

Last edited: