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Topology Final
| Term | Definition |
|---|---|
| metric | (non-degeneracy) if d(x,y)=0 iff x=y (symmetry) if d(x,y) = d(y,x) for all x,y elements X (triangle inequality) d(x,z) < d(x,y) + d(y,z) for all x, y, z in X |
| topology | 1. X, empty set are in T 2. closed under arbitrary unions 3. closed under finite intersections |
| basis | (covering) for all x in X, there is a B in B such that x is in B (refinement) for x is in B1 intersect B2, where each Bi is in B, then there is a B3 in B such that x is in B3 with is contained in B1 intersect B2 |
| Hausdorff | Whenever x does not equal y, are two distinct points of X, there are open neighborhoods U of x and V of y such that U and V are disjoint |
| continuous | In a function f: X --> Y, for every open set U of Y, we have that f-1(U) is open in X |
| diameter | diam(A) = sup{d(x,y) | x, y in A} |
| open | In a function between f: X ---> Y, for every open set U of X, the image f(U) is open in Y |
| converges | For a sequence xn, a point x in X, for every neighborhood U containing x, there is an N in the natural numbers such that for all n > N, we have that xn in U. |
| Pasting Lemma | If X = A union B where A and B are closed, and we have two continuous functions f: A ---> Y and g: B ---> Y such that f=g on A intersect B, then there is a continuous function h: X ---> Y such that h|a = f and h|b = g. |
| closure | the intersection of all closed sets containing A |
| interior | the union of all open subsets of X contained entirely with int(A) |
Created by:
kan91830
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