By Edwin E. Moise (auth.)
Geometric topology might approximately be defined because the department of the topology of manifolds which offers with questions of the life of homeomorphisms. merely in particularly contemporary years has this type of topology accomplished a sufficiently excessive improvement to take delivery of a reputation, yet its beginnings are effortless to spot. the 1st vintage consequence was once the SchOnflies theorem (1910), which asserts that each 1-sphere within the airplane is the boundary of a 2-cell. within the following couple of a long time, the main striking affirmative effects have been the "Schonflies theorem" for polyhedral 2-spheres in area, proved through J. W. Alexander [Ad, and the triangulation theorem for 2-manifolds, proved through T. Rad6 [Rd. however the such a lot extraordinary result of the Twenties have been detrimental. In 1921 Louis Antoine [A ] released a rare paper within which he four confirmed number of believable conjectures within the topology of 3-space have been fake. therefore, a (topological) Cantor set in 3-space needn't have a easily attached supplement; for this reason a Cantor set may be imbedded in 3-space in at the least basically other ways; a topological 2-sphere in 3-space don't need to be the boundary of a 3-cell; given disjoint 2-spheres in 3-space, there isn't inevitably any 3rd 2-sphere which separates them from each other in 3-space; and so forth and on. the well known "horned sphere" of Alexander [A ] seemed quickly thereafter.
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Extra resources for Geometric Topology in Dimensions 2 and 3
Sphere in R2 . Then R2 - J is not connected. PROOF. Let i be a polyhedral 2-cell containing J, such that J n Fr I contains exactly two points P and R. ) Then J is the union of two arcs A 1 and A 2, from P toR. Take a broken line B, from S to Q, in i, and intersecting Fr I only at S and Q. Let T be the first point of B (in the order from S to Q) which lies in J; let A 1 be the arc from P to R in J that contains T; and let A 2 be the other arc from P to R in J. 2 Lemma 1. A 2 contains a point of B, following X in the order from S to Q on B.
Now take any a 2 , and subdivide the edges so as to get a 1-dimensional complex with the same number of edges as L(v). Form a subdivision of a 2, using the vertices of the subdivision of Fr a 2 , and using one new vertex which lies in no edge of a 2 . 1 0( b). 10 38 4 The Jordan curve theorem Theorem 9. Let K be a complex, such that M = IKI is a 2-manifold with boundary. Then K is a combinatoria/2-manifold with boundary, and Bd M is the union of the edges of K that lie in only one 2-simplex of K.
_Let X be a topological space and let U be an open set. Then Fr U= U- U. By definition, Fr U = U n X- U. Therefore Fr U e U. Since U is open, we have U n X- U = 0. Since Fr U eX- U, it follows that Fr U e U- U. Next observe that if P E U- U, then P E U and P EX- U eX- U. Therefore U- U e Fr U. The theorem follows. 0 PROOF. Theorem 5. Let J be a polygon in R2 , with interior I and exterior E. Then every point of J is a limit point both of I and of E. PROOF. Let F = Fr I= j- I. Then F separates R2 : R2 - F = I U (R2 - i ), and the sets on the right are disjoint, open, and nonempty; R2 - i contains e J, and F is closed.