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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?

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    $\begingroup$ It is a stronger form of Hausdorff. $\endgroup$ Commented Jun 28, 2019 at 1:12

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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 $h(b)=1$.

So the condition you're looking at here is a generalization of how in a metric space, we can "walk away" from a point nicely. As often happens with these, its primary use is that it replaces having a metric structure (or being metrizable) as a sufficient hypothesis in some theorems, while being a weaker condition overall; that is, it lets you extend some theorems about metric spaces to more complicated (non-metrizable) topological spaces.

Incidentally, spaces where every pair of points are separated by a continuous function are called completely Hausdorff.

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