The question is pretty much in the title, I'm looking for an example of a locally connected space and continuous mapping such that the image is not locally connected.
Thanks!
EDIT: Corrected the phrasing to the intended meaning.
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$\begingroup$ Did you want some condition on the image of the map? Otherwise, pick any non-locally connected space and any map from a single point to this space. $\endgroup$– Dan RustJun 25, 2013 at 15:40
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6$\begingroup$ The identity from $\Bbb R$ with the discrete topology to the Sorgenfry Line. $\endgroup$– David MitraJun 25, 2013 at 15:42
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$\begingroup$ @James: The image $f(U)$ need not be open unless the map $f$ is an open mapping. $\endgroup$– Stefan HamckeJun 25, 2013 at 15:46
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1$\begingroup$ See my answer to the question Is the image of a path or arc locally path/arc connected?. It shows that even the continuous image of the unit interval need not be locally connected. $\endgroup$– Stefan HamckeJun 25, 2013 at 15:53
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$\begingroup$ May be the following will work. Let $X=[0;1]$ and $f:X\to\mathbb C$ such that $f(x)=xe^{i/x}$ for each $x\in X$. $\endgroup$– Alex RavskyJun 25, 2013 at 15:55
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4 Answers
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Boring example: any space $X$ is the continuous image of the discrete topology on $X$ (using the identity and noting that any function with a discrete domain is continuous). A discrete space is trivially locally connected (all singleton sets). Now let $X$ be any non-locally connected space.
answered Jun 26, 2013 at 4:07
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Consider the following variant on the topologist's sine curve.
This space $X$ consists of the graph of $y = \sin(\pi/x)$ for $0<x<1$, together with a closed arc from the point $(1,0)$ to $(0,0)$. Note that $X$ is not locally connected at $(0,0)$.
However, there exists a continuous surjection $f\colon [0,2)\to X$. Specifically, $f(0) = (0,0)$ and $f(1) = (1,0)$, with $f(t)$ following along the bottom curve for $0\leq t\leq 1$. For $t>1$, the function follows along the sine curve, i.e. $$ f(t) \;=\; \left(2-t,\sin\left(\frac{\pi}{2-t}\right)\right)\qquad\text{for }t> 1. $$
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1$\begingroup$ Another nice example. I really liked that graph, is there a simple way to create such graphs? :) $\endgroup$ Jun 25, 2013 at 17:23
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$\begingroup$ I borrowed the picture from a previous answer of mine. I don't recall how I made it, but I would guess Mathematica. $\endgroup$– Jim BelkJun 25, 2013 at 18:03
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$\begingroup$ why is X not connected in (0,0)? isnt it that neighbor of X is not separable? $\endgroup$– e.adApr 4, 2022 at 8:19
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For example, consider $X = \mathbb{N}\cup \{0\}$ under discrete topology and $Y = \{0\}\cup\{\frac{1}{n}|n\in \mathbb{N}\}$ as subspace of real line $\mathbb{R}$. Define $f:X\to Y$ by $f(0)=0,f(n)=\frac{1}{n},n\in \mathbb{N}$.Then $f$ is a bijection. and $f$ is continuous. Therefore, $Y$ is continuous image of $X$. Note that $X$ is being a discrete space is locally connected but $Y$ is not locally connected.
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$\begingroup$ Welcome to math.SE! Please, learn some typesetting. Moreover, if $f:X\to Y$, why define it in $0$? $\endgroup$– user228113May 13, 2015 at 17:58
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The graph-parametrisation of the graph of $\sin \frac{1}{x}$ for $x>0$ ? Instead of letting the graph trail off to the right as $x\to\infty$, just turn it around and let the curve run along the interval $[-1,1]$ on the $y$-axis. Then every point of this interval will have the property that local connectivity fails.
answered Jun 25, 2013 at 15:54
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$\begingroup$ Maybe this is more reasonable as a comment? $\endgroup$– Dan RustJun 25, 2013 at 15:54
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1$\begingroup$ What? That graph is locally connected. $\endgroup$ Jun 25, 2013 at 15:56
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1$\begingroup$ It seems that the graph of $\sin \frac{1}{x}$ for $x>0$ is homeomorhic to $\mathbb R$. $\endgroup$ Jun 25, 2013 at 15:57
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$\begingroup$ I added the part of the curve along the $y$-axis. $\endgroup$ Jun 26, 2013 at 7:17