There are objects that have different shapes but nevertheless have the same topological dimension. How can you get the topological dimension of an object?
A good Gift-Book for understanding Topology with concepts : https://au.mathworks.com/academia/books/matlab-codes-for-finite-element-analysis-ferreira.html
There is a number of books in this subject e.g.
Geometry and Topology
CUPReid M., Szendroi B.Year:2005
Topology: Pearson New International Edition
Pearson Education LimitedMunkres, James RaymondYear:2014
A Course in Mathematical Analysis, vol. 2: Metric and Topological Spaces, Functions of a Vector Variable
CUPGarling D.J.H.Year:2014
Introduction to Metric and Topological Spaces
Oxford University PressWilson A SutherlandYear:2009
Ok thank you very much, I have a question. I can say that two objects with different shapes that present the same fractal dimension are topologically equivalent.
In topology there are several definitions of dimension. Under assumptions of the space, e.g., metric space where the topology is defined by a metric, several of the definitions overlap. But there are many types of "pathological" but important sets and spaces where the definitions of dimension do not overlap, e.g., the Cantor sets.
See for a good discussion.
http://u.cs.biu.ac.il/~megereli/final_topology.pdf
A 2 dimensional sphere and a two dimensional torus are both two dimensional manifolds. That is at each point there a neighborhood that is homeomorphic (in fact diffeomoorphic) to a neighbor of the origin in the two dimensional plane. However, as topological spaces there are not homeomorphic or topologically equivalent since the genus of the sphere is 0 and the torus 1. Dimension is a local property.
https://en.wikipedia.org/wiki/Genus_(mathematics)
A classic text in topology and geometry is Singer and Thorp, "Lecture Notes on Elementary Topology and Geometry." It's not free on the web.
http://webmath2.unito.it/paginepersonali/sergio.console/Dispense/Singer%20Thorpe%20Topolog%23682C8.pdf
The following are useful;
(1) V. V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I, (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.
(2) https://en.wikipedia.org/wiki/Lebesgue_covering_dimension.
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