Topology: terms w many definitions

Convergence

- lim as x approaches infinity of Xn = X <=> ∀ ε > 0, ∃ N>0 s.t. d(Xn, X) < ε ∀ n>N - for all G in T, G contains x, there exists N st xi in G i>N

Open Set

- G subset of (X,d), G open ∀ x in G, ∃ r>0 s.t. B(x,r) = {y in X | d(x,y)<r} subset of G - A is open iff A = int(A)

Limit Point

- A is a subset of (X,T). We say x in X is a limit point of A <=> ∀ G in T s.t. x in G, (G/{x}) n A is nonempty - We say X is a limit point of A: B(X, ε) n A does not equal 0 for all ε>0 - x is a limit point of A iff there exists a sequence of distinct points in A that converge to x

Dense

- A is dense iff cl(A) = X - A is dense in X iff B(x,r) n A does not equal empty set for any x in X

Relatively Open

- A subset G of Y is relatively open in Y ∃ open set U in X with G = U ∩ Y - (X,d), Y subset of X, we say G subset of Y is relatively open ∀ x in G, ∃ r s.t. B(x,r) ∩ Y is a subset of G

Boundary of A

- Cl(A) n Cl(X\A) - Cl(A)\int(A)

Closed Set

- Complement is open - A is closed iff A = cl(A) - A is closed iff A contains all its limit points - F is closed iff for any {Xn} in F such that Xn -> X, then X in F - Compact sets are closed

Complete

- Every Cauchy sequence is convergent - X complete <=> ∀ closed Fi s.t. F1 bigger than F2 bigger than F3,... And diam(Fi) → 0, then infinite ∩ of F = {single point}

Compact Set

- K is compact iff for any open cover G of K, we can always find a finite subcover - For any f in K, has FIP, we have nF in fF does not equal empty set - Every sequence in K has a convergent subsequence - Every infinite subset in K has a limit point in K - (K,d) is complete and K is totally bounded

Normal

- Normal implies regular and regular implies hausdorff - Metric or compact => normal - can separate two closed sets w disjoint open sets - A is a closed subset of open G. There exists open subset U such that A is a subset of U which is a subset of cl(U) which is a subset of G

Continuous

- Pre image of open subset is open - Pre image of closed subset is closed - f:(X,d) → (Z, p) is continuous: <=> ∀ x in X and ∀ ε>0, ∃ δ>0 s.t. d(y,x)< δ => d(f(y),f(x))< ε - f continuous <=> lim Xn = X => lim f(Xn) = f(X)

Equivalent Metrics

- The identity map is a homeomorphism (X,d) -> (X, p) - We say d and p are equivalent metrics <=> ∃ A,B > 0 s.t. Bp(x,y) ≤ d(x,y) ≤ Ap(x,y)

Int(A)

- U G s.t. G open and G subset of A - largest open set containing A - X\cl(X\A)

Connected Set

- X is connected iff X has no open and closed subsets other than empty set and X - X = AuB (A, B disjoint), A, B both open and closed => A = empty set and B = X - E in R. E connectef iff E is an interval

Regular

- disjoint closed set and point can be separated by two disjoint open sets - Compact or metric space - For all x in G open, there exists an open subset U st x in U a subset of cl(U) a subset of G - F is a closed subset, x in X\F, then there exists an open subset V st F is a subset of V and x is not in cl(V)

Uniformly Continuous

- ∀ ε>0, ∃ δ>0 s.t. d(x,y)< δ => d(f(x),f(y))< ε - ∀ ε>0, ∃ δ>0 s.t. B(X, δ) is a subset of f-1(B(f(X), ε)) for all x in X

Cl(A)

- ∩F s.t. F closed and A subset of F - smallest closed set containing A - X\int(X\A) - A U limit points of A - {x in X such that dist(x, A) = 0}