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Show If a point A is in the interior, then it has a neighborhood contained in A.

  • Thread starter Hodgey8806
  • Start date
  • #1
145
3

Homework Statement


Let A be a topological space and let A[itex]\subseteq[/itex]X be any subset.
Show: If a point A is in the interior, then it has a neighborhood contained in A.


Homework Equations


Neighborhoods are defined to be open in my book.
Int(A) = [itex]\bigcup[/itex]{C[itex]\subseteq[/itex]A and C is open in X}


The Attempt at a Solution


Let p[itex]\in[/itex]Int(A).
Then p[itex]\in[/itex][itex]\bigcup[/itex]{C[itex]\subseteq[/itex]X:C[itex]\subseteq[/itex]A and C is open in X}
So, [itex]\exists[/itex] an open set C' s.t. p[itex]\in[/itex]C' and C[itex]\subseteq[/itex]A
Q.E.D.
 

Answers and Replies

  • #2
22,097
3,280
Sounds good!
 
  • #3
145
3
Thank you! Could you check my work backwards now?
I can say:

Let C'[itex]\subseteq[/itex]A be open and non-empty
Let p[itex]\in[/itex]C
Thus p[itex]\in[/itex][itex]\bigcup[/itex]{C[itex]\subseteq[/itex]X:X[itex]\subseteq[/itex]A and C is open in X}
Thus p[itex]\in[/itex]Int A
 
  • #4
22,097
3,280
That's correct. But you got to clean up your presentation. What's C' for example??

Thank you! Could you check my work backwards now?
I can say:

Let C'[itex]\subseteq[/itex]A be open and non-empty
Let p[itex]\in[/itex]C
Thus p[itex]\in[/itex][itex]\bigcup[/itex]{C[itex]\subseteq[/itex]X:X[itex]\subseteq[/itex]A and C is open in X}
Thus p[itex]\in[/itex]Int A
 
  • #5
145
3
C' was meant to be one set that has that element and
I meant to type let p[itex]\in[/itex]C'
and I meant to write
p∈⋃{C⊆X:C⊆A and C is open in X}

Do I still need to clean up?
 
  • #6
22,097
3,280
Ah ok. I guess it's ok now!
 
  • #7
145
3
Thank you very much! If you have time, would you mind helping me with my other question? It's about the equivalence of a bounded set with a closed ball. I just want to check my proof one direction. Thanks again!
 

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