Continuous functions: Images and Pre-images of closed and open sets

I'm having a bit of difficulty seeing whether a function or its inverse preserves whether a set's image/pre-image remains open of closed. There are 5 parts for which I must either prove or give a counter-example. I cannot find an example for **3.** and can't think of how to prove **4. **I put all of them down though - just in case I messed anything up.

**Setup**:

Let be continuous. Also, let and

**1.** If is open, then is open.

False: If for some constant , then its range is closed no matter the pre-image.

**2.** If is closed, then is closed.

False: Using and , then .

**3.** If is bounded, then is bounded.

False: __But__ I'm having difficulty thinking of an example.

**4.** If is open, then is open.

True: __But__ again, I'm not sure where to turn for the proof.

**5.** If is closed, then is closed.

True: If I had the above, I'd use its complement to show this one holds, but since I don't... Let and let be a sequence of elements from that converges to . Since is continuous, converges to . Since is closed, is in C. So, is in .

Re: Continuous functions: Images and Pre-images of closed and open sets

I realize I can use the complement of **5.** to get **4.** - I would, however, still appreciate a direct method if anyone knows.

As for **3.**, I might be going crazy .. intuitively, I think the statement is true. After all, since is continuous on , it is continuous on . If it were an interval l could look at , where and , I could use continuity to show that the function is bounded on the interval. But is a set - and I'm not sure how that changes things.

Re: Continuous functions: Images and Pre-images of closed and open sets

Quote:

Originally Posted by

**jsndacruz** **Setup**:

Let

be continuous. Also, let

and

**1.** If

is open, then

is open.

False: If

for some constant

, then its range is closed no matter the pre-image.

**2.** If

is closed, then

is closed.

False: Using

and

, then . **3.** If

is bounded, then

is bounded.

False:

__But__ I'm having difficulty thinking of an example.

**4.** If

is open, then

is open.

True:

__But__ again, I'm not sure where to turn for the proof.

**5.** If

is closed, then

is closed.

True: If I had the above, I'd use its complement to show this one holds, but since I don't... Let

and let

be a sequence of elements from

that converges to

. Since

is continuous,

converges to

. Since

is closed,

is in C. So,

is in

.

Be careful in #2. is not close. But is closed, while is not.

Quote:

Originally Posted by

**jsndacruz** I realize I can use the complement of

**5.** to get

**4.** - I would, however, still appreciate a direct method if anyone knows.

As for

**3.**, I might be going crazy .. intuitively, I think the statement is true. After all, since

is continuous on

, it is continuous on

. If it were an interval l could look at

, where

and

, I could use continuity to show that the function is bounded on the interval. But

is a set - and I'm not sure how that changes things.

Every bounded set of real numbers is the subset of a closed finite interval.

So it is the subset of a compact set. The function is continuous. Thus?

Re: Continuous functions: Images and Pre-images of closed and open sets

Ah - for #2 I meant to use square brackets. Thanks for paying attention to me!

My professor never taught us about 'compact' sets - my grasp on topology is really poor. But I think I know what you're getting at: since S is a subset of a closed interval, it is sufficient to show that the function is bounded on that interval, since the subset of a bounded set must be bounded.