# Give $[0,1]^{\omega}$ the uniform topology. Find an infinite subset of this space that has no limit point.

Give $[0,1]^{\omega}$ the uniform topology. Find an infinite subset of this space that has no limit point.

I know there are many post about this question but I would like to understand this solution that I found on the internet, could someone help me please?

Interesting. $A=\{x_i=(0,...,0,1,0,...)\}$ has no limit point (if $x$ has a coordinate in $(0,1)$ then a small enough ball does not contain any other point in $A$ , otherwise the distance to any point in $A$ is either $0$ or $1$).

I do not understand this: (if $x$ has a coordinate in $(0,1)$ then a small enough ball does not contain any other point in $A$ , otherwise the distance to any point in $A$ is either $0$ or $1$)

• What is your definition of "uniform topology"? – user223391 Nov 19 '17 at 3:25
• @ZacharySelk It is the topology generated by the uniform metric. – user402543 Nov 19 '17 at 3:26

Note that $(e_i)$ is an infinite subset of $[0,1]^\omega$ which has no limit point because $d(e_i,e_j)=\sup|e_i-e_j|=1$ where $e_i$ has $1$ in the $i^{th}$ co-ordinate and $0's$ elsewhere.
The uniform metric is given by $d(\overline x,\overline y)=\sup_{i\in I} |x_i-y_i|$ where $\overline x=(x_i),\overline y=(y_i)$
Since $e_i,e_j$ differ in atleast one co-ordinate so $d(e_i,e_j)=1$
If the sequence $(e_i)_i$ has to have a limit point then after a finite number of terms distance between any two terms must be very very small(which is what the definition of limit point says) which is not here as $d(e_i,e_j)=1$ for $i\neq j$