Somewhere dense is defined to be the following:
Let B, t be a topological space and C ⊂ B. C is somewhere dense if (Cl C)o ≠Ø, the empty set. That is, if the closure of the interior of C has at least one non-empty set.
See related links for more information.
ref
A discrete topology on the integers, Z, is defined by letting every subset of Z be open If that is true then Z is a discrete topological space and it is equipped with a discrete topology. Now is it compact? We know that a discrete space is compact if and only if it is finite. Clearly Z is not finite, so the answer is no. If you picked a finite field such a Z7 ( integers mod 7) then the answer would be yes.
No. By definition, planes can be extended in all directions to infinity. If they are not parallel, they will intersect somewhere.
A cone with included interior is 3-dimensional. However, if you are not including the interior it is a 2-dimensional surface residing in a 3-d ambient space. If you're utilizing the common topological definition of dimension, you can derive that a cone (surface only) is 2 dimensional by looking at its open sets.
The comb space C=([0,1]X0) union (KX([0,1]) union (0X{0X1]} where K is the set 1/n where n is an integer. It is made up of vertical lines that make it look like a comb. Each of these vertical lines is joined at the bottom to the y axis. You can see immediately the C is connected since each vertical segment is connected and each vertical segment meets the horizontal segment which is also clearly connected. Now, we need to show it is NOT locally connected. Note the following are equivalent: (TFAE) 1. A space X is locally connected 2. Components of open subsets in X are open ( in X) 3. X has a basis consisting of connected subsets Let V be an open ball with the usual metric in the comb space, which I will call C. Let's put V at the point (0,1/2) and the ball has radius 1/4. The vertical segments of the comb will be the components of V. All of these are open except for ones along the y axis. So we have the {0,y| which is an element of R2 1/4<y<3/4} is not open. This violates condition 2 and we have C is not locally connected. Note the comb space is path connected as is the deleted comb space. But the comb space is not path connected.
Negative space is the space where something can be seen that was not necessarily created. These are added effects in paintings for example.
Let B, t be a topological space and let C ⊂ B. The interior of C, written Co is the union of all of the open sets within C. This can be expressed using set theory notation asCo = ∪{P Є t | P ⊂ C}.See related links for more information.
In mathematics, a zero-dimensional topological space is a topological space that ... any point in the space is contained in exactly one open set of this refinement.
a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of Aan open set is an abstract concept generalizing the idea of an open interval in the real line
you put the sub in space!
A topological space is simply a set, B, with topology t (see the related link for a definition), and is often denoted as B, t which is similar to how a metric space is often denoted; B, D.
A Betti number is a number associated to each topological space and dimension, giving an approximate number of holes of that dimension in that space.
no
-adjective, dens⋅er, dens⋅est. 1. having the component parts closely compacted together; crowded or compact: a dense forest; dense population. 2. stupid; slow-witted; dull. 3. intense; extreme: dense ignorance. 4. relatively opaque; transmitting little light, as a photographic negative, optical glass, or color. 5. difficult to understand or follow because of being closely packed with ideas or complexities of style: a dense philosophical essay. 6. Mathematics. of or pertaining to a subset of a topological space in which every neighborhood of every point in the space contains at least one point of the subset.
Vector spaces can be formed of vector subspaces.
A family of subsets of the direct product of a topological space with itself that is used to derive a uniform topology for the space. Also known as uniform structure.
The wikipedia article says, 'The definition of a topological space relies only uponset theory and is the most general notion of a mathematical "space" that allows for the definition of concepts such as continuity, connectedness, and convergence.'These are abstract spaces where distance is, in some sense, ignored. When Euler considered the 'seven bridges of Koenigsberg problem', for instance, he appreciated that he was ignoring the distances between the bridges and was considering only how they were connected--so that someone could traverse each of them just once. Since that time, of course, the idea of a topological space has permeated many areas of mathematics.See the related link.
Determine the nullspace/kernel of the contrability matrix