# Convergence in Hausdorff vs. non-Hausdorff spaces

In Rudin's functional analysis text, he defines convergence in Hausdorff spaces in the following way:

$$x_n\rightarrow x$$ if every neighborhood of $$x$$ contains all but finitely many $$x_n$$.

Is this the definition for convergence in topological spaces generally? If so, any idea why he would specify Hausdorff spaces? I mean I supposed the definition is only clearly meaningful in Hausdorff spaces since this kind of bunching in neighborhoods of $$x$$ is only 'special' in a space in which we are capable of separating points into disjoint neighborhoods. Is that the only thing going on here? Is there a better way to understand convergence in non-Hausdorff spaces?

• I think Rudin makes the distinction because in Hausdorff spaces, the limit will be guaranteed to be unique. In non-Hausdorff spaces, it may be that every sequence with infinitely many distinct points "converges" to every point in the space (think of $\Bbb R$ with the cofinite topology). – Clayton Mar 19 '19 at 13:33
• The definition of $x_n\to x$ works in any topological space, Hausdorff or not. But people often use the alternative notation $\lim_{n\to\infty}x_n=x$, and this notation should be restricted to Hausdorff spaces. In non-Hausdorff spaces, you could have $\lim_{n\to\infty}x_n=x$ and $\lim_{n\to\infty}x_n=y$ and yet $x\neq y$, contrary to how the symbol "$=$" should behave. – Andreas Blass Mar 19 '19 at 19:17

This is the definition of convergence of sequences in general topological spaces, however sequences aren't always sufficient when talking about convergence. The usual definition for convergence of a sequence in general topological spaces is the following:

Definition: A sequence $$(x_{n})$$ in a topological space $$X$$ is said to converge to a point $$x\in X$$, if for every open neighbourhood $$U\subseteq X$$ of $$x$$ there is an $$N\in\mathbb{N}$$ such that for every $$m\geq N$$ one has that $$x_{m}\in U$$.

This is exactly the definition you gave in your question. However, if your topological space is not Hausdorff, then a sequence may converge to multiple points. For example let $$X$$ be the underlying set of $$\mathbb{R}$$ and equip it with the indiscrete topology. That is, the only open sets are $$\mathbb{R}$$ and $$\emptyset$$. Then given any sequence $$(X_{n})$$ in $$X$$ we have that $$(x_{n})$$ converges to every point of $$X$$.

If $$X$$ is assumed to be Hausdorff then a sequence can converge to at most one point.

Prop: Let $$X$$ be a Hausdorff space and $$(x_{n})$$ a sequence that converges to the points $$x$$ and $$y$$. Then $$x=y$$.

If $$x\neq y$$ then there are disjoint open sets $$U,V\subseteq X$$ that contain $$x$$ and $$y$$ respectively, say $$x\in U$$. Because $$(x_{n})$$ converges to $$x$$ we have that $$U$$ contains all but finitely many of the $$x_{n}$$. This implies that there are infinitely many of the $$x_{n}$$ that are not in $$V$$, an open neighbourhood of $$y$$. This contradicts $$(x_{n})$$ converging to $$y$$. Therefore $$x$$ and $$y$$ must be the same.

Interestingly enough, convergence of sequences is, in general, insufficient when talking about limits. By this I mean that if $$X$$ is a topological space and $$x\in X$$ is a limit point of a set $$A\subseteq X$$ then you can not say that there is a sequence $$(x_{n})$$ in $$A$$ that converges in $$X$$ to $$x$$. An easy example is the the set $$X=\omega_{1}+1$$, the set of all countable ordinials together with the first uncountable ordinal, equipped with the order topology. The element $$\omega_{1}$$ is a limit point of the set of countable orders in this topology, but there is no sequence of countable ordinals in $$X$$ that converges to $$\omega_{1}$$.

Even worse (better) there are spaces where limits of ordinal indexed sequences are not enough. I believe that the token example is the Tychonoff plank in which there is a an element that is in the closure of a set, but no ordinal indexed sequence in that set converges to that point.

The most general notion of convergence within a space is that of the convergence of nets. In a topological space $$X$$, a point $$x\in X$$ is a limit point of a set $$A\subseteq X$$ if and only if there is a net in $$A$$ that converges to $$x$$. Instead of adding more definitions to this already long post, I will refer you to wikipedia.