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I have a question. Let $(X,\tau)$ be a topological space. Let $X_d$ be the discrete topology and $X_c$ the coarse topology on $X$.

Let $$I: X_d \to X_c$$

be the identity function between this topologies.

First of all, how can you set an identity function between two different topological spaces? Where do you send, for example, the open set $\{a,b\}$ of $X_d$ if we suppose that $X=\{a,b,c\}$?

And why are these functions nor open nor closed? I understand why there are continuous, but, in the way I understand $I$, it sends every set to the vacuum except the whole space $X$, so it always send opens to opens hence $I$ is open.

Thanks for your help!

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  • 1
    $\begingroup$ The identity function is between the set $X$ and itself, regardless of the topology: it sends each element onto itself. So for example the element $a$ is sent onto element $a$, therefore the open set $\{a\}$ has image the set $\{a\}$, which is not open. (why?) $\endgroup$ – Riccardo Orlando Nov 4 '16 at 17:51
  • $\begingroup$ God, how silly... it's friday evening and my mind knows it. Thank you, post it as an answer and I'll accept it! $\endgroup$ – Relure Nov 4 '16 at 17:52
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The identity function is between the set $X$ and itself, regardless of the topology: it sends each element onto itself. So for example the element $a$ is sent onto element $a$, therefore the open set $\{a\}$ has image the set $\{a\}$, which is not open, as long as $X$ has more than one element.

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The identity map is between set so for every $x\in X_d$, $I(x)=x\in X_c$. It is not open because it maps the open set $\{x\}$ in the set $\{x\}$ which is not open in $X_c$ unless $X$ is a single point.

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