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| Question | Answer |
| Definition 4.1: Analytic definition of continuous/cts | A 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 |
| Definition 4.2: Continuous/cts with pre-images | Given 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. |
| Theorem 4.2 | If \(\underline{B}\) is a basis for space Y, then f:X⟶Y is cts iff f-1(B) is open in X ∀ B ∈ \(\underline{B}\). |
| Theorem 4.5 | Let 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)) |
| Continuity at a Point | Define 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. |
| Theorem 4.6 | A function f:X⟶Y is cts iff f is cts at every x∈X. |
| Theorem 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 |
| Theorem 4.9 | A 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)) ) |
| Definition 4.14: Homeomorphisms Notation: X≅Y | A 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 |
| Definition: Open Map | Any 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. |
| Homeomorphism 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. |
| Definition: Embedding | A function f:X⟶Y is an embedding if f makes X homeomorphic to the subspace f(x) of Y. |
| Definition: Topological Properties | A 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) |
| Homeomorphism Fact | A 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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