I am currently writing my thesis and basically the conclusion is, in part, a statement about how this work can be built upon. I received a post-doc that is almost a natural extension of my thesis. One of the reasons I accepted it was so I could work on some ideas that I have had that I never...
I think so but the proof then seems a little trivial
By definition if the topology has every set of X it is a discrete topology. Since it is a discrete topology the compliment to each subset must be in the topology and therefore every set is closed.
Anyways thanks for your help
I think so but the proof then seems a little trivial
By definition if the topology has every set of X it is a discrete topology. Since it is a discrete topology it must include all compliments in the topology and thus every set
So the actual problem is
"Let (X,T) be a topological space with the property that every subset is closed. Prove that it is a discrete space."
So the example is just to show that I can create an arbitrary topology with all closed sets that is not a discrete space. Thus, I have come up with a...
"Let X be any non-empty set and let T be the collection of all subsets of X. Then T is called the discrete topology on the set X . The topological space (X,T ) is called a discrete space"
Then is goes on to show
"If (X,T ) is a topological space such that for every x ∈ X, the singleton set...
Yeah, but I am not sure how this proves it is a discrete space as the example I gave above is all closed (They are open sets they are referred to in the text as "clopen" by the definition) but it does not contain any singletons. So given that the above example consists of all closed sets that...
So here is the question as written, "Let (X,T) be a topological space with the property that every subset is closed. Prove that it is a discrete space."
The definition of a closed set in the book is given by:
" Let (X,T) be a topological space. A subset S of X is said to be a closed set in...
I am trying to learn some topology and was looking at a problem in the back of the book asking to show that a topological space with the property that all set are closed is a discrete space which, as understand it, means that all possible subsets are in the topology and since all subsets are...
Thank you for your comment. For most of my models I use sklearn in python or keras for ANN , occasionally I will use R. Matlab is probably my favorite language when solving most engineering problems in course work, Image processing, and the small of amount of dynamic models I work on. I have yet...
Hi all,
I just want to gain some perspective as I am sure there are people in the same boat or have been where I am.
I am in the home stretch of my Ph.D in Biomedical Engineering, I have manuscript published and a couple more in the pipeline. I work in big data and how to use it to come up...
So the book I am reading is "A Primer on Reproducing Kernel Hilbert Spaces". So they initially talk about extrinsic vs intrinsic topology on a finite topology, ##\mathbb R^n##. they claim that ##\mathbf V \subset \mathbb R^n## endowed with an inner product. They say
" the configuration involves...
I have been reading a lot about Reproducing Kernel Hilbert Spaces mainly because of their application in machine learning. I do not have a formal background in topology, took linear algebra as an undergrad but mainly have encountered things such as, inner product, norm, vector space...
Yes. Ultimately this is a regression problem. I have observed values x and observed values Y and want to know the mutual information between them with the knowledge that Y is a linear function of x. For example let's say I send a signal x which is received by a receiver that transforms x by...