Exercise · ZW-CONTINUOUS-MAP-TOPOLOGICAL-007 · ★★☆ Intermediate
If \(f:X\to Y\) and \(g:Y\to Z\) are continuous, what can be said about \(g\circ f\)?
Choose one option.
A It is continuous only if \(f\) is bijective B It is continuous only if \(g\) is injective C It need not be continuous D It is continuous, since \((g\circ f)^{-1}(W)=f^{-1}\big(g^{-1}(W)\big)\) Show answer D. It is continuous, since \((g\circ f)^{-1}(W)=f^{-1}\big(g^{-1}(W)\big)\)← Back to the question
Explanation If \(W\) is open in \(Z\), \(g^{-1}(W)\) is open in \(Y\), and then \(f^{-1}(g^{-1}(W))\) is open in \(X\). No extra hypothesis is needed.
A function \(f:X\to Y\) between topological spaces \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) is continuous if the preimage of every open set \(V\) of \(Y\) is an open set of \(X\).
\[V\in\mathcal{T}_Y\ \Longrightarrow\ f^{-1}(V)\in\mathcal{T}_X\]
More practice problems: continuous function between topological spaces ZW-CONTINUOUS-MAP-TOPOLOGICAL-001 Let X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\ and Y=\a,b\ with T=\∅,\a\,\a,b\\. Which function f:X→ Y…ZW-CONTINUOUS-MAP-TOPOLOGICAL-002 Let X=\1,2,3\ with T=\∅,\2\,\1,2\,\2,3\,\1,2,3\\ and Y=\a,b,c\ with T=\∅,\b\,\a,b\,\b,c\,\a,b,c\\.…ZW-CONTINUOUS-MAP-TOPOLOGICAL-003 What must a function f:X→ Y between topological spaces satisfy to be continuous?ZW-CONTINUOUS-MAP-TOPOLOGICAL-004 Let X be a set with at least two points. Which statement is true?ZW-CONTINUOUS-MAP-TOPOLOGICAL-005 In usual ℝ, which of these functions f:ℝ→ℝ is continuous?ZW-CONTINUOUS-MAP-TOPOLOGICAL-006 Let f:ℝ→ℝ, f(x)=x^2, with the usual topology. What is f^-1big((1,4)big)?ZW-CONTINUOUS-MAP-TOPOLOGICAL-008 Let X=\1,2,3\ have the trivial topology and Y=\a,b\ have T=\∅,\a\,Y\. How many continuous functions…ZW-CONTINUOUS-MAP-TOPOLOGICAL-009 With X=\1,2,3\, T_X=\∅,\1\,\1,2\,X\ and Y=\a,b\, T_Y=\∅,\a\,Y\, how many functions X→ Y are continu…