# What is the intuition and motivation behind separating, with a continuous function, two points in a topological space?

Suppose that $$(X,\mathcal T)$$ is a topological space, and fix two distinct points $$x,y \in X$$. We say that these two points can be separated by a continuous function is there exists a continuous function $$f:X\overset{}{\rightarrow}[0,1]$$ such that $$f(x)=0$$ and $$f(y)=1$$, where $$[0,1]$$ is a subspace of $$\Bbb R$$ with it's usual topology.

What is the motivation behind such a function? What is it's use, and if any, what is the intuition behind it?

• It is a stronger form of Hausdorff. Commented Jun 28, 2019 at 1:12

Suppose we're in a metric space. Then there's an easy way to separate any two (distinct) points $$a,b$$ with a continuous function:
• First, consider the function $$f(x)=d(x,a)$$. We have $$f(a)=0$$ and $$f(b)\not=0$$.
• Now "normalize" and look at the function $$g(x)={d(x,a)\over d(a,b)}$$. This has $$g(a)=0$$ and $$g(b)=1$$.
• Finally, we "cap" the range and let $$h(x)=\max\{1, {d(x,a)\over d(a,b)}\}.$$ And this $$h$$ is easily checked to be continuous, and has $$h(a)=0$$ and $$h(b)=1$$.