In general, in definition of manifold we have supposed the topological space is Hausdorff and second countable. Some books also explained that there is no need to suppose these two conditions. The example is also present that non-Hausdorff manifold as "the line with two origin". Then what problem will occur in higher dimensional manifolds if we suppose non-Hausdorff and non-second countable topological space.
Also, why we are taking topological space in the definition of manifold. If we take a general space then what problem will occur?
Please see answer your question on page 53 of the book "Differential Geometry of Manifolds" by Prof. Uday Chand De and Prof Absos Ali Shaikh published by Alpha Science International Ltd.
Dear Colleague
Read these two books.
Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4
Dear Pankaj,
A topology is needed so that one can talk about continuous functions.
A manifold is supposed to generalize the concepts of a curve and a surface, both of which are Hausdorff. The prototype of a manifold is the Euclidean space, which is also Hausdorff.
Non-Hausdorff spaces are very important in algebraic geometry. Algebraic varieties with the Zariski topology are non-Hausdorff. However, they are not manifolds.
The second countability condition is there mainly to guarantee the existence of a smooth partition of unity.
For the second countable condition, see the following example. Such an example does not satisfy this condition while other conditions in the definition of manifold still hold:
The full set is R^2. Define the open neighbourhood of the point (x,y) as {(z,y)|a
There is no "need" to assume these conditions. In a free country, you are free to define whatever you want. However, there are very concrete reasons to assume these two properties, and if you give up on these conditions you lose something:
(1) The Hausdorff condition ensures that limits of sequences are unique. If you do not assume it that you cannot talk about THE limit of a sequence, but only about A limit. This makes analysis on such spaces very technical.
(2) Second countability is less important for the elementary theory of manifolds, but it is crucial for most of the more adcanced analysis, since it guarantees the existence of a partition of unity, and hence of global objects like Riemannian metrics. It is also crucial for the study of function spaces on manifolds, since it gives separability of such spaces.
In which subarea of topology you want to study? The problem arise if that subarea need Hausdorffness and second countability. For general topological study you may not need these.
Just a short addendum to Tobias Hartnicks answer: Second countability of a manifold measures in a way that your manifold does not get "too big". Meaning that it admits an embedding into some euclidean space. However, for many of the standard constructions (i.e. Riemannian metrics as outlined in Tobias post) it suffices to ask for paracompactness. In particular, paracompactness is usually the prerequesit used in the local to global approach as it allows patching together things by partition of unity.
Note however that this is only mildly more general as for a finite-dimensional paracompact manifold each connected component is second countable.
if you need independent local neighbourhood then you need Hausdorfness. Second countability need if you want to take local things to global aspects and that done through partition of unity and there you need also Hausdorffness. See differential manifold by Lee, Boothby, A R Sharsry etc etc
Dear Pankaj,
a/ If the topology is not Hausdorff then the convergence sequences will not have unique limit point.(this is necessary and sufficient condition to have a unique limit point.
b/ First countability allow to use ordinary sequences instead generalized (The compactness and sequential compactness coincide if and only if the topological space is first countable)
c/ Second countability is first countable and implies several fundamental advantages connected with metrizable properties:
1. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent;
2. Every Hausdorff regular second-countable space is metrizable.
For example for linear topological space a linear manifold is not necessary to contain the zero element of the LTP.
The given set is associated a unique topology. This topology may not be Hausdorff.
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