- Thread starter
- Banned
- #1

- Thread starter Poirot
- Start date

- Thread starter
- Banned
- #1

If $X$ is a metric space, then $X-\emptyset$ is open, then $X$ is closed. Also, $X-\emptyset$ is closed so $X$ is open. So the whole set is both open and closed.Are all metric spaces open sets? How can this be proved?

Last edited:

- Thread starter
- Banned
- #3

It is a metric space ie a Topological space. What are the requirements for a topological space?how do you know X without the empty set is closed?

- Thread starter
- Banned
- #5

The statement that a set $O$ is open means that if $x\in O$ then some ball $\mathcal{B}(x;\delta)\subseteq O$.I'm learning about metric spaces before I go on to topological spaces. I thought there would be a proof independent of that.

By definition that is true for $\forall x\in X$.

It is vacuously true for $\emptyset$.

So the whole space and the emptyset are open.

- Thread starter
- Banned
- #7

- Feb 29, 2012

- 342

$(i) \quad \; \; d(x,y) \geq 0;$

$(ii) \quad \; d(x,y) = 0 \iff x=y;$

$(iii) \quad d(x,y) = d(y,x);$

$(iv) \quad \, d(x,y) \leq d(x,w) + d(w,y),$

where $x,y,z \in X$.

- Thread starter
- Banned
- #9

You are mixing up definitions.Well, if the definition doesn't stipulate that X is open, I don't quite see how X is open 'by definition', as Plato claims.

A metric space is a pair: a set and a metric defined on that set.

Open sets are then defined in terms of that metric.

Thus the space must be open by definition.

A set is open if each of its points is in a

Well every

So once again, here is how it works.

We start with a pair: $(X,d)$.

Define a

Then define what it means to say a set is open.

- Thread starter
- Banned
- #11