Discrete topology and discrete subspaces

[gottfried]

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?

 

[Dick]

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.

 

[gottfried]

Thanks for confirming. Very hard to be sure since there always seems to be a catch.