"A continuous function preserves closeness of points. A discontinuous function maps arbitrarily close points to points that are not close. The precise definition of continuity involves the relation of distance between pairs of points. […] continuity, a property of functions that allows stretching, shrinking, and folding, but preserves the closeness relation among points."
"Intuitively, two spaces that are homeomorphic have the same
general shape in spite of possible deformations of distance and angle. Thus, if
two spaces are not homeomorphic, they will tend to look distinctly different.
Our job is to specify the difference. To do this rigorously, we need to define
some property of topological spaces and show that the property is preserved
under transformations by any homeomorphism. Then if one space has the property
and the other one does not have the property, there is no way they can be homeomorphic."
"Numerical invariants and invariant properties enable us to distinguish certain topological spaces. We can go further and associate with a topological space a set having an algebraic structure. The fundamental group is the most basic of such possibilities. It not only provides a useful invariant for topological spaces, but the algebraic operation of multiplication defined for this group reflects the global structure of the space."
"Topology is the study of geometric objects as they are transformed by continuous deformations. To a topologist the general shape of the objects is of more importance than distance, size, or angle."
"The definition of homeomorphism was motivated by the idea of
preserving the general shape or configuration of a geometric figure. Since path
components are significant characteristics of a space, it is certainly
reasonable that a homeomorphism will preserve the decomposition of a space into
path components. […] Suppose we are given two geometric figures that we suspect
are not topologically equivalent. If both of the figures are path-connected,
counting components will not distinguish the spaces. However, we might be able
to remove a special subset of one of the figures and count the number of
components of the remainder. If no comparable set can be removed from the other
space to leave the same number of components, we will then know that the two
spaces are not homeomorphic."
"The classification theorems of mathematics are among the
ultimate triumphs of human intellectual achievement. A classification theorem
provides a complete list of all objects in a given category as well as a scheme
for matching an unknown object from the category with exactly one of the
canonical examples."
"The Simplicial Approximation Theorem is a concise statement
of the general result for functions between any two triangulated spaces. It
says that on a suitable subdivision of the domain, any continuous function can
be homotopically deformed by an arbitrarily small amount so that the modified
function sends vertices to vertices and is linear on each edge, face,
tetrahedron, and higher-dimensional cell of the triangulation."
"The triangle inequality is perhaps the most important
property for proving theorems involving distance. The name is appropriate because
the triangle inequality is an abstraction of the property that the sum of the
lengths of two sides of a triangle must be at least as large as the length of
the third side."
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