## General Topology – Balanced Word to Balanced (Sturmian?)

Let $$E$$ to be a balanced finite word: for two subwords $$U, V$$ the same length, the number of $$1$$& # 39; sin $$U$$ differs from the number of $$1$$& # 39; sin $$V$$ by at most one.

• can $$E$$ to be pursued to an infinite balanced sequence?
• can $$E$$ to be pursued until an infinite sequence of Sturmian?

This came when I tried to solve an exercise, where I have to show that the Sturmian sequences are dense in the space of balanced sequences (with respect to the usual topology of symbolic dynamics).