what does $[0,1]^\omega$ mean

Question

The question is,

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

I just want to know what does $[0,1]^\omega$ mean so I can proceed.

I'd appreciate any explanation.

Answer 1

$[0,1]^\omega$ is defined as the set of sequences $(x_1, x_2, \dots)$ with $x_j \in [0,1]$.

The uniform topology in $[0,1]^\omega$ is defined as:

$x = \{x_i : i < \omega\}$

$y = \{y_j : j < \omega\}$

$p(x,y) = \sup\{\overline{d} (x_i, y_i) : i < \omega\}$

(Just taking it from this post, which seems to be dealing with a similar problem)

Answer 2

$[0,1]^{\omega}$ is the set of infinite sequences of elements of $[0,1]$, i.e. the set of all sequences $$(x_0, x_1, x_2, \cdots)$$ such that $x_n \in [0,1]$ for all $n$.