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What does topologically equivalent mean?
Two metrics on the same set are said to be topologically equivalent of they have the same open sets. So if an open subset, U contained in M is open with respect to one metric if and only if it is open with respect to the other metric. Another way to think of this is two objects are topologically equivalent if one object can be continuously deformed to the other. To be more precise, a homemorphism, f, between two topological spaces is a continuous bijective map with a continuos inverse. If such a map exists between two spaces, the are topologically equivalent.
What is the set of all points?
Space
Prove that comb space in a topological space is an example for connectednes but not locally connectedness?
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.
What is the set of all possible points in geometry?
Space
A boundless three dimensional set of all points?
Space
Why is every singleton set in a discrete metric space open?
In a metric space, a set is open if for any element of the set we can find an open ball about it that is contained in the set. Well for the singletons in the discrete space, every other element is said to have a distance away of 1. So we can make a ball about the singleton of radius 1/2 ... this ball just equals that singleton since it contains only that element. So it is contained in the set. Thus the singleton set is open.
When is a metric on a set complete?
A metric on a set is complete if every Cauchy sequence in the corresponding metric space they form converges to a point of the set in question. The metric space itself is called a complete metric space. See related links for more information.
What is a discrete set?
A set which is made up only of isolated points is called a discrete set.
What does fraction of discrete set of objects?
A discrete set of objects has no ability to do anything other than being a discrete set of objects. The same applies to a fraction of such a set.
What do yo mean by Discrete mathematics?
Discrete Mathematics is mathematics that deals with discrete objects and operations, often using computable and/or iterative methods. It is usually opposed to continuous mathematics (e.g. classical calculus). Discreteness here refers to a property of subjects of discourse. Some collection of things is called discrete if these things are distinguishable and not continuously transformable into each other. An example would be the collection of natural numbers, but not the real numbers. In topology, a space is called discrete if every subset is open. In constructivism, a set is called discrete if equality of two elements is always decidable.
What is a example of discrete data?
Discrete data are observations on a variable that which take values from a discrete set.
When we plot all the points that satisfy an equation or inequality what do we do?
We identify a set of points in the relevant space which are part of the solution set of the equation or inequality. The space may have any number of dimensions, the solution set may be contiguous or in discrete "blobs".
What is a continuous and discrete data set?
time to learn a song for 4 hours, is this discrete or continuous data set?
What do you call a set of numbers with an exact number of points?
This is called a discrete set (all points isolated) or a finite set. Finite sets are always discrete.
What does discrete function mean?
Discrete Function - A function that is defined only for a set of numbers that can be listed, such as the set of whole numbers or the set of integers. Explicit Definition - A definition of a function by a formula in terms of the variable.
Is a point is a zero dimensional?
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.
What notation is used to symbolize a topological 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.
