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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/05%3A_New_Page/5.02%3A_New_Pagen this section, we will introduce the notions of open, closed, compact, and connected as they pertain to subsets of the real numbers. These properties form the underpinnings of a branch of mathematics...n this section, we will introduce the notions of open, closed, compact, and connected as they pertain to subsets of the real numbers. These properties form the underpinnings of a branch of mathematics called topology (derived from the Greek words tópos, meaning ‘place, location’, and ology, meaning ‘study of’). Topology, sometimes called “rubber sheet geometry," is concerned with properties of spaces that are invariant under any continuous deformation.
- https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/1%3A_Vector_Basics/1.3%3A_2D_LimitsEquivalently, the limit is L if for all paths that lead to P, the function also tends towards P. (Recall that for the one variable case we needed to check only the path from the left and from the r...Equivalently, the limit is L if for all paths that lead to P, the function also tends towards P. (Recall that for the one variable case we needed to check only the path from the left and from the right.) To show that a limit does not exist at a point, it is necessary to demonstration that two paths that both lead to P such that f(x,y) tends towards different values.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)Motivated by questions in cosmology, the open-content text Geometry with an Introduction to Cosmic Topology uses Mobius transformations to develop hyperbolic, elliptic, and Euclidean geometry - three ...Motivated by questions in cosmology, the open-content text Geometry with an Introduction to Cosmic Topology uses Mobius transformations to develop hyperbolic, elliptic, and Euclidean geometry - three possibilities for the global geometry of the universe. The text, written for students who have taken vector calculus, also explores the interplay between the shape of a space and the type of geometry it admits.
