# The Unapologetic Mathematician

## Topology

Well, I’m not quite done with the updates of the topics, but I’ve gotten a number of other things done on my break. Now there’s a search bar over on the right, and the WordPress bug for subtopics has been handled. Rather than delay any longer, I guess I should jump back into the thick of it.

Topology is, roughly speaking, the study of spaces where we have an idea of what it means for points to be “close” to each other, and functions which “preserve closeness”. We don’t care about anything but the most general notion of shape. There’s the famous example of a coffee mug and a doughnut being “the same” to a topologist because they both have one hole, and if you make them out of clay you can deform one into the other without making any drastic changes like a sharp cut. In fact, it’s common to say that topology is all about situations like this, where our shapes are made from clay or rubber sheets that can be deformed around, but as we’ll see there are plenty of situations where we can make cuts (as long as we sew them up again nicely) or even weirder things can happen. Deformations are a good intuition for some aspects of topology, but they’re definitely not the most general.

Okay, so how can we get a handle on this notion of “closeness”. The usual way is to take the set of points $X$ we’re looking at and define some collection $T\subseteq P(X)$ of its subsets as the “open” subsets. Such a collection is required to satisfy a few rules:

• The empty set $\varnothing$ and the whole set $X$ are both in $T$
• The union $\bigcup\limits_\alpha U_\alpha$ of any collection $\{U_\alpha\}\subseteq T$ of subsets in $T$ is again in $T$
• The intersection $\bigcap\limits_{i=1}^nU_i$ of any finite collection $\{U_i\}\subseteq T$ of subsets in $T$ is again in $T$

We call the specified collection $T$ a “topology” on the set $X$, and pair of a set $X$ and a topology $T$ on $X$ we call a “topological space. The elements of $T$ we call the open sets of $X$, and their complements in $X$ we call the closed sets.

Notice here that the collection of closed sets is completely determined by the collection of open sets. This leads to an alternate viewpoint, where we define a collection $T\subseteq P(X)$ of subsets of $X$ satisfying:

• The empty set $\varnothing$ and the whole set $X$ are both in $T$
• The intersection $\bigcap\limits_\alpha U_\alpha$ of any collection $\{U_\alpha\}\subseteq T$ of subsets in $T$ is again in $T$
• The union $\bigcup\limits_{i=1}^nU_i$ of any finite collection $\{U_i\}\subseteq T$ of subsets in $T$ is again in $T$

Now the elements of $T$ are called the closed subsets of the topological space, and their complements are called the open subsets.

We can put more than one topology on the same set $X$, and we can compare different topologies. Let’s say that we have topologies $T_1$ and $T_2$ on a set $X$, so that $T_1\subseteq T_2$. That is, every subset of $X$ that $T_1$ calls open, $T_2$ does as well. In this case, we say that the topology $T_1$ is “coarser” than $T_2$, or that $T_2$ is “finer” than $T_1$. Since we define this relationship by restricting subset containment from $P(P(X))$ to those collections of subsets of $X$ which are actually topologies, it defines a partial order on the collection $\mathcal{T}$ of all topologies on $X$.

The coarsest possible topology is $\{\varnothing,X\}$, which says that only the empty subset and the whole set are open. We call this the “trivial” or the “indiscrete” topology on $X$. Conversely, the finest possible topology is $P(X)$, which says that every subset is open. This we call the “discrete” topology on $X$. Useful topologies tend to fall somewhere between these two extremes, but at least we know that $\mathcal{T}$ has a top and a bottom element for the coarseness relation.

In the middle, let’s say we have some collection $\{T_\alpha\}\subset\mathcal{T}$ of topologies. Then we can define their intersection $\bigcap\limits_\alpha T_\alpha$ as subsets of $P(X)$. This will also be a topology, as is easily shown from the definition above. It is the finest topology which is coarser than all the topologies in $\{T_\alpha\}$, and so any subset of $\mathcal{T}$ has a greatest lower bound.

On the other hand, the union of this collection may not be a topology, which could serve as a least upper bound. However, there is always at least one topology that contains this union — the discrete topology. So we can consider the collection — known to be nonempty — of all topologies which contain the union $\bigcup\limits_\alpha T_\alpha$. The intersection of this collection of topologies will be a topology (as above) which is finer than each topology $T_\alpha$, and is the coarsest possible such topology. Thus any subset of $\mathcal{T}$ has a least upper bound.

Together, these results say that the $\mathcal{T}$ is a complete lattice under the coarseness relation. This turns out to be useful when we have some set we want to put a topology on, and we want to do it in the coarsest possible way subject to a collection of requirements. The fact that $\mathcal{T}$ is a complete lattice says that we can find the coarsest possible topology satisfying the relations one at a time, and then we can find the coarsest topology finer than each of them.

November 5, 2007 - Posted by | Point-Set Topology, Topology

## 18 Comments »

1. […] as Categories Okay, so we’ve defined a topology on a set . But we also love categories, so we want to see this in terms of categories. And, indeed, […]

Pingback by Topologies as Categories « The Unapologetic Mathematician | November 9, 2007 | Reply

2. […] Okay, we know what a topological space is. As we might expect, these will be the objects of some category . So we need morphisms to […]

Pingback by Continuous Maps « The Unapologetic Mathematician | November 12, 2007 | Reply

3. […] and Closures When we pick a topology on a set , not every subset is open, and not every subset is closed. However, we can still come up […]

Pingback by Interiors and Closures « The Unapologetic Mathematician | November 13, 2007 | Reply

4. […] As we go on, we’re going to want to focus right in on a point in a topological space . We’re interested in the subsets of in which we could “wiggle around a bit” […]

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5. […] Now let’s add a little more structure to our topological spaces. We can use a topology on a set to talk about which points are “close” to a subset. Now […]

Pingback by Uniform Spaces « The Unapologetic Mathematician | November 23, 2007 | Reply

6. […] Nick Bornak under topology   I’m afraid I’ve been sucked in to all this talk of topology. Instead of going forward with my plans to broach internal set theory, I think it’s prudent […]

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7. […] of Topological Spaces We’ve defined topological spaces and continuous maps between them. Together these give us a category . We’d like to understand […]

Pingback by Limits of Topological Spaces « The Unapologetic Mathematician | November 26, 2007 | Reply

8. […] Now I want to toss out a few assumptions that, if they happen to hold for a topological space, will often simplify our work. There are a lot of these, and the ones that I’ll mention […]

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9. […] series as if the were an honest basis. The core idea is that we’re going to introduce a topology on the space of […]

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10. […] spaces. There are many other ones which give rise to different distance functions, but the same topology and the same uniform structure. And often it’s the topology that we’ll be most […]

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11. […] the ring of complex-valued functions on a domain . Instead of defining this topology in terms of open sets as we usually do, we define this topology in terms of which nets converge to which points. In fact, […]

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12. […] As we move towards multivariable calculus, we’re going to primarily be concerned with the topological spaces (for various values of ) just as in calculus we were primarily concerned with the topological […]

Pingback by The Topology of Higher-Dimensional Real Spaces « The Unapologetic Mathematician | September 15, 2009 | Reply

13. You are getting close to the answers. Keep searching. Seek out Tyler Trefoil. Comment by Incongruous | November 20, 2009 | Reply

14. Excuse me? Comment by John Armstrong | November 21, 2009 | Reply

15. […] a particular collection of “special” subsets, a measurable space should remind us of a topological space, and like topological spaces they form a category. Remember that our original definition of a […]

Pingback by Measurable Spaces, Measure Spaces, and Measurable Functions « The Unapologetic Mathematician | April 26, 2010 | Reply

16. […] so to get a topological vector space, we take a vector space and put a (surprise!) topology on it. But not just any topology will do: Remember that every point in a vector space looks pretty […]

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17. […] Remember that we defined measurable functions in terms of inverse images, like we did for topological spaces. So it should be no surprise that we move a lot of measurable structure around between spaces by […]

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18. […] of study in differential topology and differential geometry is a “manifold”. This is a topological space, which looks “close-up” like a real vector space. In fancier language, a manifold is […]

Pingback by Manifolds « The Unapologetic Mathematician | February 22, 2011 | Reply