Let f : V mapped to W be a continuous function between two topological spaces V...

Let f : V mapped to W be a continuous function between two topological spaces V and W, so that (by definition) the preimage under f of every open set in W is open in V : Y is open in W implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that the preimage under f of every closed set in W is closed in V . Feel free to take V = W = R^n to simplify things. Hint: show that the “preimage of” operation plays nice with set-complements, and then use the fact that every closed set is the complement of some open set. Note that R^n is both open and closed as a subset of itself.

Expert Solution

Colby Messinger answered 2 years ago

A function f : X ------> Y between two topological spaces ( X , TX )...

A function f : X ------> Y between two topological spaces ( X , TX ) and ( Y , TY ) is called a homeomorphism it has the following properties: a) f is a bijection (one - to- one and onto ) b) f is continuous c) the inverse fucntion f -1 is continuous ( f is open mapping) A function with these three properties is sometimes called bicontinuous . if such a function exists, we say X and Y...

1. For a map f : V ?? W between vector spaces V and W to...

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Let Vand W be vector spaces over F, and let B( V, W) be the set...

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Questionnnnnnn a. Let V and W be vector spaces and T : V → W a...

Questionnnnnnn a. Let V and W be vector spaces and T : V → W a linear transformation. If {T(v1), . . . T(vn)} is linearly independent in W, show that {v1, . . . vn} is linearly independent in V . b. Define similar matrices c Let A1, A2 and A3 be n × n matrices. Show that if A1 is similar to A2 and A2 is similar to A3, then A1 is similar to A3. d. Show that...

Let V and W be Banach spaces and suppose T : V → W is a...

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Let V and W be finite dimensional vector spaces over a field F with dimF(V )...

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Question 1. Let V and W be finite dimensional vector spaces over a field F with...

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1.- let(X1, τ1) and (X2, τ2) are two compact topological spaces. Prove that their topological product...

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4. Verify that the Cartesian product V × W of two vector spaces V and W...

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1. Let V and W be vector spaces over R. a) Show that if T: V...

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