Could you explain how to show 7.3 in detail? Definition. Let X and Y be topological spaces. A function or map f : X → Y is acontinuous function or continuous map if and only if for every open set U in Y,f-'(U) is open in X.Theorem 7.1. Let X and Y be topological spaces, and let f:Xthe following are equivalent:→ Y be a function. Then(1) The function f is continuous.(2) For every closed set K in Y, the inverse image f-l(K) is closed in X.(3) For every limit point p of a set A in X, the image f(p) belongs to f(A).(4) For every x EX and open set V containing f(x), there is an open set U containing xsuch that f(U) V.Theorem 7.2. Let X,Y be topological spaces and yo E Y. The constant map f :X → Ydefined by f(x) = yo is continuous.Theorem 7.3. Let X C Y be topological spaces. The inclusion map i : X →by i(x) = x is continuous.Y defined

Name: Could you explain how to show 7.3 in detail? Definition. Let X and Y b
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: Could you explain how to show 7.3 in detail? Definition. Let X and Y be topological spaces. A function or map f : X → Y is acontinuous function or continuous map if and only if for every open set U in Y,f-'(U) is open in X.Theorem 7.1. Let X and Y be