Discrete topology and discrete subspaces

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SUMMARY

The discussion centers on the relationship between discrete topology in a subspace A and the larger space X. It is established that if A has a discrete topology, it does not necessarily imply that X also has a discrete topology, as demonstrated by the example of the natural numbers as a subspace of the real numbers with the Euclidean metric. Conversely, if X has a discrete topology, then every singleton in X must be open, which implies that any subspace A will also be discrete. The criteria for a set A to be open in X is confirmed as requiring that for all elements a in A, the distance d(a, X - A) must be greater than zero.

PREREQUISITES
  • Understanding of discrete topology and its properties
  • Familiarity with metric spaces and the Euclidean metric
  • Knowledge of open sets and their definitions in topology
  • Basic concepts of subspaces in topology
NEXT STEPS
  • Study the properties of discrete topology in various metric spaces
  • Explore examples of subspaces and their topological properties
  • Learn about open sets in topology and their implications
  • Investigate the implications of singleton sets in metric spaces
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Mathematicians, students of topology, and anyone interested in understanding the nuances of discrete topology and its implications in metric spaces.

gottfried
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Homework Statement


If A is a subspace of X and A has discrete topology does X have discrete topolgy?Also if X has discrete topology then does it imply that A must have discrete topology?

The Attempt at a Solution


My understanding of discrete topology suggests to me that if A is discrete it doesn't imply that X is also. example A the natural numbers as a subspace of X the real numbers with the euclidean metric.

I feel the reverse is true. For X to be discrete does every singleton in the metric space have to be open? ie: is it correct to say that the a metric topology is discrete if and only if each singleton in the metric space is open?
I think the answer is yes in which case each singlton in X is open meaning any subspace is also discrete. Since the subspace would be a collection of singletons and unions of singletons.

Also for a set to be open is true to say: a set A a subset of X is open if and only if for all a in A d(a,X-A)>0?
 
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gottfried said:

Homework Statement


If A is a subspace of X and A has discrete topology does X have discrete topolgy?Also if X has discrete topology then does it imply that A must have discrete topology?



The Attempt at a Solution


My understanding of discrete topology suggests to me that if A is discrete it doesn't imply that X is also. example A the natural numbers as a subspace of X the real numbers with the euclidean metric.

I feel the reverse is true. For X to be discrete does every singleton in the metric space have to be open? ie: is it correct to say that the a metric topology is discrete if and only if each singleton in the metric space is open?
I think the answer is yes in which case each singlton in X is open meaning any subspace is also discrete. Since the subspace would be a collection of singletons and unions of singletons.

Also for a set to be open is true to say: a set A a subset of X is open if and only if for all a in A d(a,X-A)>0?

That all seems pretty ok to me.
 
Thanks for confirming. Very hard to be sure since there always seems to be a catch.
 

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