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Show that if $X$ is path-connected and $f:X\to Y$ is a continuous map, then the image $f(X)$ is path-connected.

In order to show this is path connected I know the definition is :

Definition: A topological space $X$ is path connected if $\forall x,y \in X \, \, \exists$ continuous function $\gamma: [0,1] \rightarrow X$ such that $\gamma(0)=x$ and $\gamma(1) = y$. i.e. any two points can be connected with a continuous path.

Irregular User
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george
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2 Answers2

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Suppose $x,y\in f(X)\subset Y$ so that we can say $x=f(a)$ and $y=f(b)$. Then there is a path $\gamma:[0,1]\rightarrow X$ such that $\gamma(0)=a$ and $\gamma(1)=b$. The composition, $f\circ \gamma$ is then a continuous path on $f(X)$ with the desired property.

Eoin
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Let $y_0 = f(x_0), y_1 = f(x_1)$ two elements of $f(X)$.

Consider a path joining $x_0$ to $x_1$: $$ \gamma \in C([0, 1], X);\\ \gamma(i) = x_i\ \ \ \ (i\in \{0, 1\}) $$

Then $f \circ \gamma$ is a path joining $y_0$ and $y_1$.

mookid
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  • I know I am a bit late to ask this question but, it is the reciprocal true? Does path-connected image over a continuous map imply path-connected domain? – Daniel C. Oct 05 '23 at 11:23
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    @DanielC. take a disconnected space and take a trivial map to a singleton, it is continuous and the image is connected – Micheal Brain Hurts Dec 28 '23 at 12:11