In the last topology post, I introduced the idea of a metric space, and then used it to define open and closed sets in the space.
Today I’m going to explain what a topological space is, and what continuity means in topology.
A topological space is a set and a collection of subsets of , where the following conditions hold:
- :both the empty set and the entire set are in the set of subsets, . is going to be the thing that defines the structure of the topological space.
- : the union of collection of subsets of is also a member of .
- : the intersection of any two elements of is also a member of .
The collection is called a topology on . The members of are the open sets of the topology. The closed sets are the set complements of the members of . Finally, the elements of the topological space are called points.
The connection to metric spaces should be pretty obvious. The way we built up open and closed sets over a metric space can be used to produce topologies. The properties we worked out for the open and closed sets are exactly the properties that are required of the open and closed sets of the topology.
The idea of the topology is that it defines the structure of X. We say collection when we talk about it, because it’s not a proper set: a topology can be (and frequently is) considerably larger than what’s allowable for a set.
What it does is define the notion of nearness for the points of a set. Take three points in the set : , , and . X contains a series of open sets around each of , , and . At least conceptually, there’s a smallest open set containing each of them. Given the smallest open set around , there is a larger open set around it, and a larger open set around it. On and on, ever larger. Closeness in a topological space gets its meaning from those open sets. Take that set of increasingly large open sets around . If you get to an open set around that contains before you get to one that contains , then is closer to than is.
There are many ways to build a topology other than starting with a metric space, but that’s definitely the easiest way. One of the most important ideas in topology is the notion of continuity. In some sense, it’s the fundamental abstraction of topology. Now that we know what a topological space is, we can define what continuity means.
A function from topological space to topological space is continuous if and only if for every open set , the inverse image of on is an open set.
Of course that makes no sense unless you know what the heck an inverse image is. If C is a set of points, then the image is the set of points . The inverse image of on is the set of points .
Even with the definition, it’s a bit hard to visualize what that really means. But basically, if you’ve got an open set in , what this says is that anything that maps to that open set must also have been an open set. You can’t get an open set in using a continuous function from unless what you started with was an open set. What that’s really capturing is that there are no gaps in the function. If there were a gap, then the open spaces would no longer be open.
Think of the metric spaces idea of open sets. Imagine an open set with a cube cut out of the middle. It’s definitely not continuous. If you took a function on that open set, and its inverse image was the set with the cube cut out, then the function is not smoothly mapping from the open set to the other topological space. It’s mapping part of the open set, leaving a big ugly gap.
If you read my old posts on category theory, here’s something nifty.
The set of of topological spaces and continuous functions form a category, with the spaces as objects and continuous functions as arrows. We call this category
Aside from the interesting abstract connection, when you look at algebraic topology, it’s often easiest to talk about topological spaces using the constructs of category theory.
For example, one of the most fundamental ideas in topology is homeomorphism: a homeomorphism is a bicontinuous bijection (a bicontinuous function is a continuous function with a continuous inverse; a bijection is a bidirectional total function between sets.)
In terms of the category , a homeomorphism between topological spaces is a homomorphism between objects in
But there’s more: from the perspective of topology, any two topological spaces with a homeomorphism between them are identical. And – if you go and look at the category-theoretic definition of equality? It’s exactly the same: so if you know category theory, you get to understand topological equality for free!