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  • What is the difference between topological and metric spaces?
    While in topological spaces the notion of a neighborhood is just an abstract concept which reflects somehow the properties a "neighborhood" should have, a metric space really have some notion of "nearness" and hence, the term neighborhood somehow reflects the intuition a bit more
  • Direct Limit of Sheaves on a Noetherian Topological Space
    Combining this with your observation that every subset of a noetherian topological space is quasi-compact, now you know that you only have finitely many pieces of data to keep track of, and this should let you solve the problem
  • Not fully understanding the definition of topological invariance
    According to Wikipedia, a topological invariant is defined as a property of a topological space that remains invariant under homeomorphisms Obviously, this definition is correct, but I'm failing to see how this definition is not redundant since a homeomorphism already preserves the features of a topological space Can someone please clarify?
  • general topology - Net convergence and relation to topological space . . .
    Clearly all cofinal subnets are subnets using the inclusion map, but the converse is false My query is, if we relax axioms 1-4 above to use cofinal subnets only, what would be the kind of mathematical structure we get? Clearly, these are much weaker axioms, valid for topological spaces, but perhaps many non-topological structures as well
  • real analysis - What is an open set in a topological space . . .
    Topology is weird at first, but in the abstract setting of topology you define a topology by saying what your open sets are This makes it nearly impossible to answer your question, because what an open set is depends on the topology (it is still an ok question though) The terminology of open close makes the most intuitive sense in the euclidean topology probably
  • meaning of topology and topological space
    A topological space is just a set with a topology defined on it What 'a topology' is is a collection of subsets of your set which you have declared to be 'open'
  • general topology - Definition of locally connected topological space . . .
    A topological space is locally connected if every point admits a neighbourhood basis consisting of open connected sets To the definition given by Lee (Introduction to topological manifolds - page $92$) which sums up definitions $1$ and $2$ more "compactly" as follows;
  • The graph of a continuous function is a topological manifold
    However, regarding the third condition in the definition of a topological manifold, I don't fully understand how $ \varphi $ can be homeomorphic to an open subset of $ \mathbb {R}^n $
  • What is a topological space good for? - Mathematics Stack Exchange
    Topological spaces can also be applied to settings where it's not clear how to define a metric, or even when you can't even apply the notion of metric space at all An important example is used in algebraic geometry, one aspect of which is about studying solutions to polynomial equations
  • The role of strictness in the equivalence between $ 2 $-dimensional . . .
    In it's Frobenius Algebras and 2D Topological Quantum Field Theories J Kock defines a TQFT as a symmetric monoidal functor from the symmetric monoidal category $ \mathrm {Bord} (2) $ of $ 2 $ -dimensional bordisms (or its strictification) to the symmetric monoidal category $ \mathrm {Vect}_ {\mathbb F} $ of vector spaces over the field





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