In some ways, the opposite of the discrete topology is the trivial topology ( also called the " indiscrete topology " ), which has the fewest possible open sets ( just the empty set and the space itself ).
32.
But on one of your notes, the indiscrete space is not locally metrizable ( when it has more than one point as someone above mentioned ) but I would expect that the derivative of a function between such indiscrete spaes to be 0.
33.
But on one of your notes, the indiscrete space is not locally metrizable ( when it has more than one point as someone above mentioned ) but I would expect that the derivative of a function between such indiscrete spaes to be 0.
34.
So, now, Washington being Washington, it's atwitter over an indiscrete e-mail, which was leaked to an enterprising cyber-gossip, from the wife of the White House speechwriter who apparently coined the brilliant / overwrought ( pick one ) phrase.
35.
The functor from "'Set "'to "'Cat "'that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects . ( For the right adjoint, see indiscrete category .)
36.
Yet, there is no guarantee that every point of the locale & Omega; ( " X " ) is in one-to-one correspondence to a point of the topological space " X " ( consider again the indiscrete topology, for which the open set lattice has only one " point " ).
37.
If f : X \ to \ mathbb { R } is an arbitrary non-constant, real-valued function, then f is non-measurable if X is equipped with the indiscrete algebra \ Sigma = \ { \ emptyset, X \ }, since the preimage of any point in the range is some proper, nonempty subset of X, and therefore does not lie in \ Sigma.
38.
For example, if a space is locally Euclidean at a point you can define its tangent space at that point ( which is, itself, a useful thing to do for all kinds of purposes ), you can't define the tangent space to a figure of 8 at that central point ( it has two tangents there, so you would end up with the union of two lines, which isn't a vector space ) . 3 ) A manifold is a space that is locally Euclidean everywhere ( possibly with some extra conditions, depending on who you ask ), that is the minimal condition ( I'm not really sure what you mean by " some sort " of manifold, there are generalisations of manifolds, but they are really manifolds any more even if the word may appear in their name, you could argue that " topological space " is a generalisation of " manifold ", but that doesn't mean much ) . 4 ) No, the only neighbourhood of any point in an indiscrete space is the whole space, which can't be homeomorphic to any Euclidean space because no Euclidean space ( beyond "'R "'0, I guess ) is indiscrete .-- talk ) 00 : 04, 12 January 2009 ( UTC)
39.
For example, if a space is locally Euclidean at a point you can define its tangent space at that point ( which is, itself, a useful thing to do for all kinds of purposes ), you can't define the tangent space to a figure of 8 at that central point ( it has two tangents there, so you would end up with the union of two lines, which isn't a vector space ) . 3 ) A manifold is a space that is locally Euclidean everywhere ( possibly with some extra conditions, depending on who you ask ), that is the minimal condition ( I'm not really sure what you mean by " some sort " of manifold, there are generalisations of manifolds, but they are really manifolds any more even if the word may appear in their name, you could argue that " topological space " is a generalisation of " manifold ", but that doesn't mean much ) . 4 ) No, the only neighbourhood of any point in an indiscrete space is the whole space, which can't be homeomorphic to any Euclidean space because no Euclidean space ( beyond "'R "'0, I guess ) is indiscrete .-- talk ) 00 : 04, 12 January 2009 ( UTC)
