From continuity to homeomorphism, compactness in domain

In summary, the conversation discusses whether a claim is true or not. The claim states that if X and Y are topological spaces and all closed subsets of X are compact, then all continuous bijections from X to Y are homeomorphisms. The speaker mentions a useful theorem that supports this claim and provides a proof. It is mentioned that X being compact and Y being Hausdorff is necessary for the claim to hold.
  • #1
jostpuur
2,116
19
Is this claim true? Assume that [itex]X,Y[/itex] are topological spaces, and that all closed subsets of [itex]X[/itex] are compact. Then all continuous bijections [itex]f:X\to Y[/itex] are homeomorphisms.

It looks true on my notebook, but I don't have a reference, and I don't trust my skills. Just checking.
 
Physics news on Phys.org
  • #2
There is this useful theorem that says "If X is compact and Y is Hausdorff, then any continuous bijection f:X-->Y is a homeo" and the proof goes like this:

to show: f(C) is closed for all C in X closed. Take such a C. Since X is compact, C is compact, so f(C) is compact. But compact sets in a Hausdroff and closed, QED.

Observe that to say that all closed subsets of X are compact is equivalent to saying that X is compact. So, modulo Y being Hausdorff, your claim is the above theorem.
 
  • #3
I see. I must have used the Hausdorff assumption without noticing it.
 

FAQ: From continuity to homeomorphism, compactness in domain

1. What is continuity?

Continuity is a mathematical concept that describes the smoothness and connectedness of a function. A function is continuous if small changes in the input result in small changes in the output.

2. How is continuity related to homeomorphism?

Homeomorphism is a type of continuous function that is also bijective, meaning it has a one-to-one correspondence between its inputs and outputs. This means that a homeomorphism preserves the continuity of the function.

3. What is compactness in domain?

Compactness in domain refers to a mathematical property of a set, where every open cover of the set has a finite subcover. In simpler terms, this means that the set does not have any "gaps" or "holes" and can be covered by a finite number of smaller sets.

4. Why is compactness in domain important?

Compactness in domain is important in mathematics because it allows for easier analysis and manipulation of functions. It also has many applications in other fields, such as physics and economics.

5. How is compactness in domain used in real-world problems?

Compactness in domain is often used in real-world problems to find optimal solutions or to determine the existence of solutions. For example, in economics, compactness in domain can be used to prove the existence of an equilibrium point in a market system.

Similar threads

I Is the Projection Restriction to a Linear Subspace a Homeomorphism?
Replies
8
Views
1K
B Is the Quotient Topology of Real Numbers Homeomorphic to Real Numbers?
Replies
6
Views
2K
I Is [0,1] under the subspace topology Hausdorff, Compact, or Connected?
Replies
15
Views
2K
I Homotopy Definitions: Homeomorphisms, Homotopies & Retracts
Replies
13
Views
3K
I On two definitions of locally compact
Replies
5
Views
956
Continuous Bijection f:X->X not a Homeo.
Replies
9
Views
13K
I Compact support functions and law of a random variable
Replies
11
Views
624
I Proving a function f is continuous given A U B = X
Replies
5
Views
2K
MHB Proving Relatively Compactness in C([a, b]) using Arzelá-Ascoli Theorem
Replies
2
Views
1K
I Can a Topological Space be Both a 1-Manifold and an n-Manifold?
Replies
17
Views
2K

Hot Threads

Recent Insights

Back
Top