• Support PF! Buy your school textbooks, materials and every day products Here!

Prove that [itex]f: X \rightarrow Y[/itex] is a continuous function.

  • #1
My question is:

Let [itex]f:\bigcup_{\alpha}A_{\alpha} \rightarrow Y[/itex] be a function between the topological spaces Y and [itex]X=\bigcup_{\alpha}A_{\alpha}[/itex]. Suppose that [itex]f|A_{\alpha}[/itex] is a continuous function for every [itex]\alpha[/itex] and that [itex]{A_{\alpha}}[/itex] is locally finite collection. Suppose that [itex]A_{\alpha}[/itex] is closed for every [itex]\alpha[/itex].

Show that: [itex]f[/itex] is continuous.

Any hints?

I'm stuck with this problem for some days. Some gave me answers on mathematics stackexchange. but it didn't make much sense.
 

Answers and Replies

  • #2
22,097
3,282
What definition of continuity would you use?
 
  • #3
What definition of continuity would you use?
ِA function [itex]f:X \rightarrow Y[/itex] between two topological spaces [itex]X[/itex] and [itex]Y[/itex] is continuous if the preimage of any open set of [itex]Y[/itex] is an open set of [itex]X[/itex].
 
  • #4
22,097
3,282
ِA function [itex]f:X \rightarrow Y[/itex] between two topological spaces [itex]X[/itex] and [itex]Y[/itex] is continuous if the preimage of any open set of [itex]Y[/itex] is an open set of [itex]X[/itex].
Yes. Do you perhaps know of others characterizations of continuity that would be more handy in this case?
 
  • #5
Sure, I know that a function is continuous if the preimage of an closed set is closed. and a set is closed iff its closure is the set itself. I lately heared that an element x is in the closure of a set if every neighborhood of [itex]x[/itex] intersects the set itself.

a function is continuous iff for every [itex]x[/itex] in the domain. the preimage of a neighborhood [itex]U[/itex] of [itex]f(x)[/itex] is a neighborhood of [itex]x[/itex].

Note: I got an answer for the question here (Mathematics StackExchange )but I still concerned in knowing different ideas for the problem as it puzzeled me for several days :)
 

Related Threads on Prove that [itex]f: X \rightarrow Y[/itex] is a continuous function.

Replies
1
Views
2K
Replies
2
Views
784
Replies
0
Views
767
Replies
1
Views
976
Replies
2
Views
4K
Replies
5
Views
2K
Replies
1
Views
758
Top