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Here is a quote from John Lee's Introduction to Topological Manifolds, Second Edition:

The definition of topological spaces is wonderfully flexible, and can be used to describe a rich assortment of concepts of "space". However, without further qualification, arbitrary topological spaces are far too general for most purposes, because they include some spaces whose behavior contradicts many of our basic spatial intuitions.

By requiring a space to be Hausdorff we exclude "pathological" topological space, such as the one in which a sequence could converge to more than one point.

Then, why do we not require all topological spaces to have the Hausdorff property? What do we gain by defining topological spaces to be more general spaces than Hausdorff spaces?

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    $\begingroup$ There are important topological spaces that are not Hausdorff. For example, most spaces in algebraic geometry are not Hausdorff $\endgroup$ – leibnewtz Oct 23 '18 at 1:52
  • $\begingroup$ Because there are good examples that aren't Hausdorff, including some nice finite $T_0$ models of important (and Hausdorff) spaces like spheres. $\endgroup$ – Randall Oct 23 '18 at 1:53
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    $\begingroup$ In the OP's defense, I've never seen a manifold text allow non-Hausdorff manifolds. So, if this was your only exposure, you'd only see arguments in favor of Hausdorff-ness. $\endgroup$ – Randall Oct 23 '18 at 1:55
  • $\begingroup$ I recently heard about physical midels of the universe that were not Hausdorff. $\endgroup$ – Behnam Esmayli Oct 23 '18 at 1:56
  • $\begingroup$ What would it even mean to "require all topological spaces to be Hausdorff"? Are you saying it should be forbidden to talk about non-Hausdorff spaces? Maybe you can do that in your country. Here in the United States we have something called the First Amendment. $\endgroup$ – bof Oct 23 '18 at 3:08
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In fact, in the early 20th century, when topology was being "invented"or "discovered" as a separate mathematical branch, there were several approaches in defining topologies in the first place, as a set with some extra structure. Kuratowski used axioms for closure (as did Čech) and also demanded that $\overline{\{x\}} = \{x\}$ in some of his texts (so $T_1$ was assumed throughout). Fréchet used convergence notions (of sequences) and also assumed that constant sequences had unique limits (so $T_1$ too). Hausdorff used an axiom system based on neighbourhood systems and assumed as one of the axioms that two distinct points had at least 2 disjoint respective neighbourhoods, the axiom that was later named after him. So in the early days people "sneaked in" low separation axioms as part of their definitions, mostly for convenience. Later the open sets/closed sets axioms developed and were quite generally found to be convenient (and similar to other structures being developed at that time, like $\sigma$-algebras etc.) and people proved the general "equivalence" of many of these approaches. At that time the separation axioms were formulated as separate assumptions.

The system with bare-bones axioms and simple extra assumptions (Trennungsaxiome like $T_0, T_1, T_2$ etc.) won out. That way we can easily tell which results need which extra assumptions etc.

And of course later many applications of non-Hausdorff spaces were found too, which helps.

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  • $\begingroup$ Thank you very much, $\endgroup$ – gladimetcampbells Oct 24 '18 at 17:10
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Other than the mentions in the comments that point out usefullness of some nin Hausdorff spaces, a second reason could be that we apply various actions on spaces such as quotienting that may result in non hausdorff offsprings.

Also adding Hausdorff to a definition of topological space would ruin the simplicity and generality of the definition we have now.

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  • $\begingroup$ Do you have any example of topological properties that would be ruined if we add the Hausdorff property to the definition of topological spaces? $\endgroup$ – gladimetcampbells Oct 23 '18 at 2:03
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    $\begingroup$ If you made Hausdorffness an axiom, I don't think we could hardly ever create quotient spaces, because now you're demanding that they be Hausdorff. $\endgroup$ – Randall Oct 23 '18 at 2:03
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For one thing, there would go the trivial (indiscrete) topology, the coarsest (or, depending on your point of view, I have heard, finest) topology on a space.

Also, there would go spaces which are only $T_0$ or $T_1$.

In short, without some more pathological spaces, and finer shades of meaning, topology would be a less rich subject.

The separation axioms are fairly natural, and build on one another.

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