Flashcards
§4 Continuous Functions and Homeomorphisms
Cards in this deck (14)
- 1Definition 4.1: Analytic definition of continuous/ctsA function f:ℝ⟶ℝ is \(\underline{continuous}\) at \(x_0\) if ∀ε>0, ∃ δ>0 s.t. ∀x, if |x-\(x_0\)|< δ, then |f(x)-f(\(x_0\))|<ε
- 2Definition 4.2: Continuous/cts with pre-imagesGiven any top. spaces X & Y, a function f:X⟶Y is called \(\underline{continuous}\) if the f-preimage of each open subset of Y is an open subset of X,
i.e. f-1(V) is open in X ∀ V open in Y. - 3Theorem 4.2If \(\underline{B}\) is a basis for space Y, then f:X⟶Y is cts iff f-1(B) is open in X ∀ B ∈ \(\underline{B}\).
- 4Theorem 4.5Let f:X⟶Y be cts.
Then for any A⊆X,
if x∈\(Cl_X\)(A),
then f(x)∈\(Cl_Y\)(f(A)).
Thus f(\(Cl_X\)(A)) ⊆ \(Cl_Y\)(f(A)) - 5Continuity at a PointDefine a function f:X⟶Y between top. spaces to be cts at x, where x∈X if,
for every nbhd U of f(x) in Y there is a nbhd V of X s.t. f(V)⊆U. - 6Theorem 4.6A function f:X⟶Y is cts iff f is cts at every x∈X.
- 7Theorem 4.8\(X \xrightarrow{f} Y\) is cts iff the pre-image of every closed subset of Y is closed in X
i.e. C closed in Y ⟹ f-1(C) closed in X - 8Theorem 4.9A composition of cts functions is cts:
if f:X⟶Y is cts & g:Y⟶Z is cts, then their composition g∘f:X⟶Z is cts.
(where (g∘f)(x) = g(f(x)) ) - 9Definition 4.14: Homeomorphisms Notation: X≅YA function f:X⟶Y is a \(\underline{homeomorphism}\) if
(1) f is bijective (=injective+surjective); &
(11) f is cts; &
(111) the inverse function f-1:Y⟶X, having f-1(y)=x iff y=f(x) is cts.
Thus f is a "bicontinuous bijection"
If there is a homeomorphism from X to Y then X is \(\underline{homeomorphic}\), or topologically equivalent to Y - 10Definition: Open MapAny f:X⟶Y satisfying,
U open in X ⟹ f(U) open in Y,
is called an \(\underline{open-map}\).
This homeomorphism is a cts and open bijection. - 11Homeomorphism Facts(1) For any space X, the identity functions \(id_X\):X⟶X is a homeomorphism. Hence X≅X.
(11) If f:X⟶Y is a homeomorphism, then f-1:Y⟶X is a homeomorphism. Hence X≅Y ⟹ Y≅X.
(111) If f:X⟶Y & g:Y⟶Z are homeomorphisms, so is g∘f:X⟶Z. Hence X≅Y & Y≅Z⟹ X≅Z. - 12Definition: EmbeddingA function f:X⟶Y is an embedding if f makes X homeomorphic to the subspace f(x) of Y.
- 13Definition: Topological PropertiesA property of space is \(\underline{topological}\) if it is preserved by homeomorphism.
This means: if X has the property & X≅Y, then Y has the property.
(e.g. being Hausdorff is topological Thm4.17) - 14Homeomorphism FactA homeomorphism f:X⟶Y preserves closures, interiors and boundaries:
f(\(Int_X\)(A))=\(Int_Y\)(f(A))
f(\(Cl_X\)(A) = \(Cl_Y\)(f(A))
f( \(δ_X\)(A)) =\(δ_Y\)(f(A))
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- Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Topology (MATH313)) Flashcards on §4 Continuous Functions and Homeomorphisms, created by Amy Wogan on 27/04/2016.
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