Topology

[holyholy]

a, b belong R³

d(a,b) = ((a₁ - b₁)² + (a₂ - b₂)² + (b₃ - b₃)²)^1/2

(R³, d) is a metric space. For x belongs R³, r > 0,
U_r(x) = { y belongs R³ : d(y,x) < r}, the open ball of radius r centered at x.

let T be the set of all subsets U of R³
if x belongs U, then there is r > 0 such that U_r(x) belongs U.
Show that (R³, T) is a topological space.

But I have no idea.
Should I get the proof from definition?

 

[pka]

Show that (R³, T) is a topological space.
Should I get the proof from definition?

Well of course. All proof is from definitions and axioms.

What properties must the set \(\displaystyle \mathcal{T}\) have in order for \(\displaystyle (\mathbb{R}^3,\mathcal{T})\) to be a topology?

Then show each of those apply to \(\displaystyle \mathcal{T}\).

 

[holyholy]

But I do not understand what 'the set of all subsets U of R³' means.
Thus, does R³ include in T?

 

[pka]

But I do not understand what 'the set of all subsets U of R³' means.
Thus, does R³ include in T?

.
Well if you do not indeed understand \(\displaystyle \mathbb{R}^3\), quite frankly you have no business trying this problem.

\(\displaystyle \mathbb{R}^3\) is the set of all triples of real numbers.

\(\displaystyle U_r[(a,b,c)=\) the set of all points in \(\displaystyle \mathbb{R}^3\) whose distance from \(\displaystyle (a,b,c)\) is less than \(\displaystyle r\).
That is known as an open ball in \(\displaystyle \mathbb{R}^3\).

The collection of all open balls along with \(\displaystyle \mathbb{R}^3~\&~\emptyset\) forms a topology.

 

[holyholy]

T is a collection of open balls where r > 0
If r -> infinity, R³ includes in T
Can I say that?
Thus, empty and R³ belongs T

finish 1 of 3 definitions

But topology seems that it is difficult to visualize.

 

[holyholy]

If it collects all open balls, the union and intersection must be included.
Because it collects all open balls at any points x.

If so, why does it provide the property?