Set of Integers: Open or Closed?

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Homework Help Overview

The discussion revolves around the classification of the set of integers in terms of topological properties, specifically whether it is open or closed within a given topology.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definitions of open and closed sets, questioning the topology being used for classification. There is a discussion about the implications of the complement of the set of integers and the nature of sets that may be neither open nor closed.

Discussion Status

Some participants have provided guidance on the definitions and properties of open and closed sets, while others are clarifying the assumptions regarding the topology in question. There is acknowledgment of a misunderstanding regarding the classification of the set, but no explicit consensus has been reached.

Contextual Notes

Participants are considering the topology inherited from the reals and discussing the implications of this choice on the classification of the set of integers.

julia89
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Homework Statement




is the set of integers open or closed

Homework Equations





The Attempt at a Solution



I thought not closed
open because R/Z=Union of open intervals
like ...U(-1,0)U(0,1)U...
 
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You got it backwards!
 
With what topology? The topology inherited from the reals? (That is, the metric topology with d(m,n)= |m- n|.)

Morphism's point is that since R\Z is a union of a union of open intervals, The complement of Z is open and so Z itself is ?

However, don't think that "open" or "closed" are all the options. It is possible for a set to be neither open nor closed. It is even possible for a set to be both open and closed.
 
ow sorry I switched open and closed
I meant that it was a closed set and not open
 
Yes, then that's right. The set is closed and not open.
 

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