From Intelligent Perception
Topology is usually defined as follows
So, that you can bend, stretch and shrink etc but not tear or glue.
This definition might be the only way to capture the essence of topology in a single sentence but this sentence is meaningless to a person who knows nothing about continuity. Meanwhile the question "Why do we need to study topology?" still remains.
So, we start with examples instead. They come from three seemingly unrelated areas: computer vision, cosmology, and data analysis.
1 Computer vision in engineering
In industrial settings one might need to consider the integrity of objects being manufactured.
The first question may be: this bolt is supposed to hold two things together, does it still or is there a crack in it?
In other words: would it hold a hair or might one slip through?
The second question may be: this sheet of material is supposed to hold liquid, is it water tight or is there leakage?
In other words: would it hold water or might it permeate through?
The third question may be: to be strong this alloy is supposed to be solid, is it or are there air bubbles?
In other words: would it hold air or might it get out?
It is important to understand now that these are three different kinds of integrity loss as there may be a crack but no hole or vice versa, etc:
We can describe these situations informally as:
Note however that the presence of these features doesn't have to be a defect. The corresponding examples of objects that are supposed to have them are: sliced bread, a strainer, and an air balloon.
Further, these three types of "defects" are so different in nature that they are associated to different dimensions. To understand why recall from linear algebra the $0$-, $1$-, and $2$-dimensional spaces: a point, a line, a plane. If they are allowed to be deformed the list becomes: points, curves, surfaces. Finally:
- The simplest object with a "cut" is two points, so this is a $0$-dimensional feature.
- The simplest object with a "tunnel" is a circle, so this is a $1$-dimensional feature.
- The simplest object with a "void" is a sphere, so this is a $2$-dimensional feature.
More on topology in computer vision: Topological features of images.
What is the shape of the Universe? What is its topology? Does the Universe have "cuts", "tunnels", or "voids"?
Looking around we don't see anything like that. But remember how, hundreds of years ago, sailors started to notice the curvature of the horizon? And later proved that the surface of the Earth encloses something inside?
As for the Universe, we know is that we are not living in the flat world of Euclid and Newton - we are living in the curved world of Riemann and Einstein. So if the Universe curves and bends, it's possible that it closes on itself!
It may be possible then to travel in a straight line and arrive to the starting point from the opposite direction. In fact, the light of the sun may come to us from an unexpected direction as if from a distant star...
If we can observe such an event, it would provide us with clues about the topology of the Universe. That would be a nice break because we can't examine it directly... Indeed, we can't step aside and take a view of the universe!
Since we live in our Universe, we can't see its "cuts", "tunnels", or "voids". The only way we can discover its structure is through travel (either ourselves or observing the light) in various directions followed by some kind of analysis. This suggests that using cycles as a tool is not just a matter of convenience but necessity.
In addition, as the universe is $3$-dimensional, it might have $3$-dimensional topological features. We can't even imagine what they look like.
We have to deal with even higher dimensions, below.
3 Data analysis
Datasets live outside of our tangible, physical, $3$-dimensional world.
Suppose we have conducted $1000$ experiments with a set of $100$ various measurements in each. Then each experiment is a vector of dimension $100$. Moreover, the complete output of the set of experiments is a "point cloud" of 1000 points in the $100$-dimensional space.
Now, as a scientist you are after patterns. So, what is the shape behind this data?
Yet we still need to answer the same questions about the object behind the point cloud:
- Is it one piece or more?
- Is there a tunnel or a void?
- And what about possible $100$-dimensional topological features?
The first question is both very important and well understood. It is about decision making and is called "clustering":
Just as important for pattern recognition than these "global" properties may be the local topology of the data. For example, in the images above the datasets are $3$-dimensional but locally they are $2$-dimensional (i.e., surfaces). This is called "dimensionality reduction".
For more on this see Topological data analysis.
Other topics in applied topology:
- topology of configuration spaces,
- folding paths of proteins,
- topology of networks,
- networks of sensors, and
- communications networks.
To summarize, one could define topology as:
It is about counting as explained in Topology, Algebra, and Geometry.