(Very) Easy Topology

**sleepingcat** · October 7th 2008, 01:15 PM

Can someone please describe the standard topology on the closed interval [0,1]? It seems (to me) like it should consist of the open balls, {0}, and {1} (and the union/finite intersection of these, etc.)... Thanks.

**Plato** · October 7th 2008, 01:28 PM

Originally Posted by sleepingcat

Can someone please describe the standard topology on the closed interval [0,1]?

If there is a standard topology on [0,1] it has countable subbasis.
So what do you mean by standard?

**ThePerfectHacker** · October 7th 2008, 01:32 PM

Originally Posted by sleepingcat

Can someone please describe the standard topology on the closed interval [0,1]? It seems (to me) like it should consist of the open balls, {0}, and {1} (and the union/finite intersection of these, etc.)... Thanks.

I assume you mean a subset topology.

Since $[0,1]$ is a subset of $\mathbb{R}$ it means $U\subseteq [0,1]$ is open iff $U = [0,1] \cap V$ for some open subset $V$ of $\mathbb{R}$ .

Therefore, the open sets of $[0,1]$ under the subspace topology are $(a,b)$ , $[0,a)$ , $(a,1]$ where $0<a<b<1$ .

**Moo** · October 7th 2008, 01:39 PM

Hello,

Originally Posted by sleepingcat

Can someone please describe the standard topology on the closed interval [0,1]? It seems (to me) like it should consist of the open balls, {0}, and {1} (and the union/finite intersection of these, etc.)... Thanks.

You don't talk about balls in this dimension oO

The standard topology of [0,1] may be the subspace topology on $[0,1] \subset \mathbb{R}$

The standard topology on [0,1] may then be $\tau=\{\mathcal{O} \cap [0,1] ~ : ~ \mathcal{O} \in \tau_{\mathbb{R}} \}$

And $\tau_{\mathbb{R}}=\{]a,b[ ~ : ~ a<b, ~a \in \mathbb{R} \cup \{-\infty\} \text{ and } b \in \mathbb{R} \cup \{+\infty\}\}$

So $\tau=\{[0,a[ ~,~ ]b,1] ~,~ ]a,b[ ~:~ 0<a \le 1 \text{ and } 0 \le b < 1 \}$

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