While revising my knowledge in general topology, I tackle various difficult problems. I came across one particular problem at which I got completely stuck. Below is the problem:
Assume: on $\mathbb{R}^{2}$ there are the set of countably many equilateral triangles $\{ T_{i}\}_{i=1}^{+\infty}$ and the set of countably many straight lines $\{ L_{i}\}_{i=1}^{+\infty}$.
Prove: there exists a point that is neither equidistant to any pair of straigth lines from $\{ L_{i}\}_{i=1}^{+\infty}$ nor a vertex of any triangle from $\{ T_{i}\}_{i=1}^{+\infty}$.
Any help will be highly appreciated!
Given two lines, the points equidistant to them is a pair of perperdicular lines. As there are countably many given lines, the set of all these pairs of perpendicular lines is also a countable collection of lines.
Hint: pick a line that is different from all of these, and see what happens on it.
(Why can you pick such a line? What is the cardinality of the points on this line? What is the cardinality of points that you cannot pick on this line?)
I would say that this problem has nothing to do with topology. It is set theory.