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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.
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Can give the Examples for compact spaces in topology?
Any closed bounded subset of a metric space is compact.
What does it mean for a subspace of a topological space to be ''somewhere dense''?
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.
In R with discrete metric space what is open set?
any interval subset of R is open and closed
What is homotopy in mathematics?
This article is about topology. For chemistry, see Homotopic groups.The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark isocontours of one possible homotopy.In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain pathological spaces. Consequently most algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
Is a cone two dimensional or three dimensional?
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.
