Thread: Basic Concepts of Topology

1. Basic Concepts of Topology

So I'm reading about sets, and if A and B are combined, their commonalities are only counted once. This makes me think of sets as being one dimensional, i.e.:

1 2 3 4 5 6 7 8 9 10

A is (1,7). B is (5,10).

Are they one dimensional?

They demonstrate sets with circles, like a Ven diagram. This confuses me.

Is it like, a three dimensional graph with X, Y, and Z. X has a set of sets, Y has a set of sets, Z has a set of sets. Then you combine these all with a matrix?

I'm also fuzzy on the concept of a neighborhood.

Are there any specific mathematical concepts that are important to master to understand topology?

2. Dimension has no meaning when you talk about sets.

Venn diagrams are a nice way to demonstrate some basic principals. Each point on the diagram is supposed to represent an element, so a set looks like a bunch of these (circle is usually used).

I have no idea what you mean with the X, Y and Z.

A neighborhood is a set which has a given point in its interior.

If you want to understand topology properly I suggest you first look at metric spaces. Topology is to metric spaces, what metric spaces are to the Euclidean setting.

3. Aha!

Topology > Metric Spaces > Euclidean Spaces

From general to specific.

When I was talking about X, Y, and Z I was thinking that each dimension of a Euclidean Space (i.e., "X") was a set... I am now under the impression that each dimension is actually a metric space, correct?

The reason dimension has no meaning when talking about sets is because dimensions are constructed by using sets?

Metric spaces are a type of set where the distance between points is specified... or simply the concept of a distance between points is possible.

How do sets translate to numbers? When understanding sets, should I stay away from visualizing them in Euclidean Space?

4. Originally Posted by lrl4565
Aha!

Topology > Metric Spaces > Euclidean Spaces

From general to specific.

When I was talking about X, Y, and Z I was thinking that each dimension of a Euclidean Space (i.e., "X") was a set... I am now under the impression that each dimension is actually a metric space, correct?
Euclidian space is the d-product of the reals with a metric on it. Strictly speaking, saying each dimension is a set is wrong as dimension is a number and not an object. The projection of the space on to a co-ordinate is a Euclidian space in its own right.

The reason dimension has no meaning when talking about sets is because dimensions are constructed by using sets?
The reason why dimension has no meaning when talking about sets is because they have no structure. The examples you keep thinking of are all canonical spaces with some sort of structure. What is the dimension of two apples, a pear and four monkeys? It just doesn't make sense.

There is also more than one definition of a dimension.

Metric spaces are a type of set where the distance between points is specified... or simply the concept of a distance between points is possible.

How do sets translate to numbers? When understanding sets, should I stay away from visualizing them in Euclidean Space?
A metric space is where distance between two elements is specified.

How you choose to visualise sets is up to you, but remember sets have no structure on their own. They translate to numbers as numbers are just sets.