Discrete topology and discrete subspaces

In summary, discrete topology suggests that if A is discrete, then X is also discrete, but the reverse is not always the case.
  • #1
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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  • #2
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.
 
  • #3
Thanks for confirming. Very hard to be sure since there always seems to be a catch.
 

1. What is discrete topology?

Discrete topology is a type of topology in which every singleton set (a set containing only one element) is an open set. This means that every individual point in a given space can be isolated and considered an open set in itself.

2. How is a discrete topology different from other topologies?

In a discrete topology, all singleton sets are open and all sets are closed. This is different from other topologies, such as the Euclidean topology, where not all singleton sets are open and closed.

3. What is a discrete subspace?

A discrete subspace is a subset of a given space that has the discrete topology. This means that all the singleton sets in the subspace are open and closed, while the remaining sets may not be.

4. Can a discrete subspace be infinite?

Yes, a discrete subspace can be infinite. The number of points in a discrete subspace is not limited, as long as each point can be isolated and considered an open set in itself. This is in contrast to other topologies, such as the finite complement topology, where the number of points is limited.

5. What are some real-world applications of discrete topology?

Discrete topology has many applications in scientific fields such as computer science, physics, and biology. It is used in computer algorithms, quantum mechanics, and the study of genetic variation, among others. It also has practical applications in data analysis and network optimization.

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