Proving f Continuous for Topological Problem on A, B Open/Closed

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In summary: No. f isn't continuous. I think you mean to say A and B don't 'intersect' not 'interact', but that still doesn't make it continuous. And f does have only one value at x=1. f(1)=2 because x=1 is in B. The problem is that x=1 is the limit of a sequence of points in x_n in A, so the value of f(x_n)=1, but f(1)=2. Look at the definition of continuity in terms of sequences again. Can you give an example of such a sequence? And why couldn't this have happened if A were closed?
  • #1
qinglong.1397
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Homework Statement


Suppose X = A[tex]\cup[/tex]B where A and B are closed sets. Suppose f : (X, TX) [tex]\rightarrow[/tex] (Y, TY ) is a map such that f|A and
f|B are continuous (where A and B have their subspace topologies). Show that f is continuous. What happens if A and B
are open? What happens if A or B is neither open nor closed?


TX means the topology on set X; TY the topology on Y. f|A means the restriction of f on A; f|B the restriction on B.

Homework Equations





The Attempt at a Solution



I do not know how to prove, so is there anyone who can give me the answer? Thank you very much!
 
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  • #2
Nobody is going to 'give you the answer'. But just think about real numbers. Suppose A=[0,1) and B=[1,2]. So X=[0,2]. Pick f(x)=1 for x in A and f(x)=2 for x in B. So f is continuous on both A and B. But it's not continuous on X. Why wouldn't this happen if A and B were closed?
 
  • #3
Dick said:
Nobody is going to 'give you the answer'. But just think about real numbers. Suppose A=[0,1) and B=[1,2]. So X=[0,2]. Pick f(x)=1 for x in A and f(x)=2 for x in B. So f is continuous on both A and B. But it's not continuous on X. Why wouldn't this happen if A and B were closed?

Thanks for your hint. But I still do not know how to generalize it. Would you please give me another hint?
 
  • #4
qinglong.1397 said:
Thanks for your hint. But I still do not know how to generalize it. Would you please give me another hint?

Not until you figure out what the last hint means. Why isn't the f I gave you continuous on X?
 
  • #5
Dick said:
Not until you figure out what the last hint means. Why isn't the f I gave you continuous on X?

OK. The reason is that A=[0, 1) does not interact with B=[1, 2], so the function f can take different values at 1. If A=[0, 1], according to the definition of function, f has only one value at 1, so it is continuous.

Am I right?
 
  • #6
qinglong.1397 said:
OK. The reason is that A=[0, 1) does not interact with B=[1, 2], so the function f can take different values at 1. If A=[0, 1], according to the definition of function, f has only one value at 1, so it is continuous.

Am I right?

No. f isn't continuous. I think you mean to say A and B don't 'intersect' not 'interact', but that still doesn't make it continuous. And f does have only one value at x=1. f(1)=2 because x=1 is in B. The problem is that x=1 is the limit of a sequence of points in x_n in A, so the value of f(x_n)=1, but f(1)=2. Look at the definition of continuity in terms of sequences again. Can you give an example of such a sequence? And why couldn't this have happened if A were closed?
 

Related to Proving f Continuous for Topological Problem on A, B Open/Closed

1. What is the definition of continuity in a topological setting?

In the context of topology, continuity refers to a function that preserves the topological structure between two spaces. This means that small changes in the input of the function result in small changes in the output, and the pre-image of an open set in the target space is an open set in the domain space.

2. How do you prove that a function is continuous in a topological setting?

To prove continuity in a topological setting, you must show that the function satisfies the definition of continuity. This typically involves using the open set definition of continuity, where the pre-image of an open set in the target space is an open set in the domain space. You may also need to use specific properties of the topological spaces involved, such as openness or closedness.

3. What is the difference between proving continuity for open sets and closed sets?

The main difference between proving continuity for open sets and closed sets is the direction of the implication in the definition of continuity. For open sets, the pre-image must be open, while for closed sets, the pre-image must be closed. This means that for open sets, the function must preserve openness, while for closed sets, the function must preserve closedness.

4. Can you give an example of a function that is continuous for open sets but not for closed sets?

Yes, a classic example of a function that is continuous for open sets but not for closed sets is the function f(x) = 1/x on the interval (0, 1]. This function is continuous for open intervals, as the pre-image of an open interval is also an open interval. However, it is not continuous for closed intervals, as the pre-image of a closed interval is not necessarily closed.

5. How are open and closed sets related to continuity in a topological setting?

In a topological setting, open and closed sets play a crucial role in defining continuity. The definition of continuity in terms of open sets means that a function is continuous if and only if the pre-image of every open set in the target space is an open set in the domain space. Similarly, the definition of continuity in terms of closed sets means that a function is continuous if and only if the pre-image of every closed set in the target space is a closed set in the domain space.

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