Question

Let $X$ be a set, $Y$ a topological space, and $X \xrightarrow{\varphi_\lambda} Y(\lambda$ in $\Lambda)$ a collection of mappings from $X$ to $Y$. Construct the coarsest topology on $X$ such that all these mappings are continuous.

   Let $X$ be a set, $Y$ a topological space, and $X \xrightarrow{\varphi_\lambda} Y(\lambda$ in $\Lambda)$ a collection of mappings from $X$ to $Y$. Construct the coarsest topology on $X$ such that all these mappings are continuous.
 
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Mathematical physics
Mathematical physics
Robert Geroch 1st Edition
Chapter 27, Problem 172 ↓

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We need to find the coarsest topology on the set \( X \) such that a given collection of mappings \( \varphi_\lambda: X \to Y \) (for each \( \lambda \in \Lambda \)) are continuous, where \( Y \) is a topological space.  Show more…

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Let $X$ be a set, $Y$ a topological space, and $X \xrightarrow{\varphi_\lambda} Y(\lambda$ in $\Lambda)$ a collection of mappings from $X$ to $Y$. Construct the coarsest topology on $X$ such that all these mappings are continuous.
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Suppose that X is a set, Y a topological space and f : X → Y a function. Show that there exists a unique weakest (coarsest) topology on X such that f is continuous.

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