You’re about to move, and you need to rent an apartment sight unseen. You go online, make some calls, and eventually settle on something that looks perfect. When you arrive at your new home, though, something’s a little…off. In fact, everything’s off. It doesn’t have electricity. You look back at the ad, and sure enough, it didn’t specifically say the apartment had power, but you didn’t think you needed to ask.
For many mathematicians, the Hausdorff property is like having power in your apartment. Of course, you can build a space without it, but you kind of assume that it will be there. Doing topology without the Hausdorff property feels like stumbling around in the dark.
The Hausdorff property, named after German mathematician Felix Hausdorff, is one of many conditions related to separation in a mathematical space: how much can points in the space be separated from each other? A space is Hausdorff if for any two distinct points in the space, you can put them in open sets that don’t intersect. Open sets are the basic units of currency in topological spaces, and you can read about why they’re important in my most recent post.
To see the importance of the Hausdorff property, think about what it means for some familiar spaces, for example, the real line. Open sets in the real line are just open intervals like (0,1). Any two points on the line, no matter how close they are, are separated by some distance, so by finding sufficiently small open intervals, you can put the points into two intervals that don’t overlap.
Other familiar spaces like the Euclidean plane or 3-dimensional space also have this property. It’s kind of hard to imagine a space that doesn’t. That’s where the line with two origins comes in. It is one of the simplest spaces that isn’t Hausdorff.
To make the line with two origins, we start with two number lines. We can label points by naming the two lines and calling the point x on the first line (x,0) and on the second line (x,1). Now we declare that the points (x,0) and (x,1) are the same point unless x=0. We also decide that the space inherits the standard topology of the real line from the two copies of the real line we started with, so open sets in this space are just open intervals.
You might object to us just deciding that those two points are the same, but doing topology often means you get to make the rules. We have always been at war with Eastasia, and (1,0) is the same as (1,1).
Can you see why the line with two origins is not Hausdorff? Unsurprisingly, it’s the “unless x=0” part of the definition that does us in. Any open interval that contains (0,0), the origin in the first number line, will overlap with any open interval that contains (0,1), the origin in the second number line because almost all the points in the first number line are identified with points in the second number line.
Although I've included a few illustrations in this post, it's hard to draw a picture of the line with two origins. In fact, it's mathematically provable that it's impossible to draw the line with two origins, and that's one of the reasons the Hausdorff property is important. The Hausdorff property doesn't guarantee that a mathematical space is easy to draw (see Cantor's leaky tent), but it does guarantee that the space can fit into some Euclidean space, perhaps a high-dimensional one, in a well-behaved way. The line with two origins can't fit in any Euclidean space in a way that really displays its essense. Instead, we have to throw a few lines and dots at the paper or computer screen and hope our audience will meet us halfway.
Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The House with Two Rooms
The Fano Plane
The Möbius Strip
The Long Line