Register to reply

Set theory/ topology question

by yxgao
Tags: theory or, topology
Share this thread:
yxgao
#1
Mar16-04, 12:39 PM
P: 124
Let f be a real-valued function defined and continuous on the set of real numbers R. Which of the following must be true of the set S = {f(c): 0<c<1}?

I. S is a connected subset of R
II. S is an open subset of R
III. S is a bounded subset of R

The answer is I and III only. I understand why I is true. But, why is is bounded, and why is it not an open subset?

Thanks.
Phys.Org News Partner Science news on Phys.org
World's largest solar boat on Greek prehistoric mission
Google searches hold key to future market crashes
Mineral magic? Common mineral capable of making and breaking bonds
matt grime
#2
Mar16-04, 01:30 PM
Sci Advisor
HW Helper
P: 9,397
it is bounded because it is continuous on the compact set [0,1] and the continuous image of a compact (closed and bounded set) is closed and bounded, the image of (0,1) is a subsert of this bounnded set and is hence bounded.

define f(x) = 0 for all x. the image of an open set is then closed.
yxgao
#3
Mar16-04, 01:56 PM
P: 124
How can you reach a conclusion by simply considering the case f(x) = 0?

matt grime
#4
Mar16-04, 04:08 PM
Sci Advisor
HW Helper
P: 9,397
Set theory/ topology question

Because the question asks if it MUST be true that the image of an open set is open. I just showed that it isn't necessarily true. To disprove a statement it suffices to provide ONE counter example.

The negation of the statement 'for all continuous f (on R) the restriction to (0,1) is an open map (ie the image is open)' is 'there exists A continuous map on r R whose restriction to (0,1) is not an open map'.


Register to reply

Related Discussions
Set Theory and Topology Mathematics Learning Materials 30
Confusion about group theory/topology notation High Energy, Nuclear, Particle Physics 2
Set theory in Munkres Topology General Math 3
Basic Set Theory/Topology Set Theory, Logic, Probability, Statistics 6
Miscellaneous pointset topology and measure theory Set Theory, Logic, Probability, Statistics 7