What is a topology? Intuition.

In summary: The phrase "the point p=0" is not a statement about the point p itself, but rather about the point 0 in the topology.
  • #1
center o bass
560
2
Hi! I'm trying to get some intuition for the notion of the topology of a set. The definition of a topology ##\tau## on a set ##X## is that ##\tau## satisfies the following:

- ##X## and ##Ø## are both elements of ##\tau##.
- Any union of sets in ##\tau## are also in ##\tau##.
- Any finite intersections of sets in ##\tau## are also in ##\tau##.

Alright, so one associates the set X with another set ##\tau## and the two together define a topological space if ##\tau## satisfies certain properties. Topology, as I understand it, is the study of the most qualitative features of a space. Have i understood it properly if i say that the role of the topology ##\tau## is to distribute the elements in ##X## in a certain way?

For example in the excluded point topology on the real line, the point ##p = 0##, is given by

$$\tau_E = \{\mathbb{R}\}\cup\{S \subseteq \mathbb{R} : 0 \notin S\}.$$

So while the point zero is in the set of reals it is not in the topology and therefore not in the topological space. Thus the topology has distributed the set of points in a certain way.

So how do you like intuitively to think about a topology and it's role?
 
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  • #2
There is an intuitive way to think about what the open sets, or the elements of the topology represent. An open set is essentially a set which 'contains all points sufficiently close to its members.'

From that idea, it is easy to see why the empty set and the space itself must be open. The space itself contains all points, so it certainly contains all the points 'sufficiently close to its members.' The empty set contains no points, and therefore there is no point in the empty set whose neighbors are also not contained in it.

Once you get the intuitive idea of what an open set is, you can use that concept to construct other concepts. For example, an 'isolated point' x does not have any neighbors, therefore the set {x} is open. You can then define an 'isolated point' as a point x such that {x} is open.
 
  • #3
It's just a type of generalization of a metric space. We lose the notion of distance, that's all. A lot of definitions from metric spaces carry over, just recast in terms of open sets instead of metrics (some things don't e.g. Cauchy sequences).
 
  • #5
The book "Modern differential geometry for physicists" by Chris Isham explains very well where the axioms of topology come from. Don't worry about the name; it's quite mathy.
 
  • #6
center o bass said:
For example in the excluded point topology on the real line, the point ##p = 0##, is given by

$$\tau_E = \{\mathbb{R}\}\cup\{S \subseteq \mathbb{R} : 0 \notin S\}.$$

So while the point zero is in the set of reals it is not in the topology and therefore not in the topological space. Thus the topology has distributed the set of points in a certain way.

So how do you like intuitively to think about a topology and it's role?


Notice that the topology is a set who's elements are sets, namely open subsets of the reals, and is not itself a subset of the reals, but of its power set.

I don't know what you mean by "the point p=0 is given by"
 
  • #7
"Topology" is essentially the study of the abstract concept of "continuity".

I don't know what you mean by "the point p=0 is given by"

He didn't say that. Look more closely. He said "excluded point topology on the real line, the point p=0 , is given by". The phrase "the point p= 0" is enclosed in commas indicating that the "excluded point" is the point 0
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric shapes and spaces that are preserved under continuous deformations, such as stretching or twisting, but not tearing or gluing.

2. What is the difference between topology and geometry?

Topology focuses on the intrinsic properties of a space, such as connectivity and continuity, while geometry focuses on the extrinsic properties, such as angles and distances.

3. How is topology used in real life?

Topology has many practical applications in fields such as physics, engineering, and computer science. It is used to analyze and design networks, study the behavior of materials, and model biological systems, among others.

4. What is the intuition behind topology?

The intuition behind topology is that it studies the fundamental structure of a space, rather than its specific shape or size. It focuses on the relationships between points and their properties, rather than the exact measurements of those points.

5. What are some common topological concepts?

Some common topological concepts include continuity, connectedness, compactness, and homeomorphism. These concepts help to classify and distinguish different types of spaces and their properties.

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