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How to describe all continuous maps from $T'$ to $T$, where $T=\mathbb{R}$ with natural topology (base given by the intervals $(a,b)$ ), and $T'=\mathbb{R}$ with the topology with basis given by the half-intervals $(a,b]$ ?

Obviously, all "linear" maps $F: (a,b] \to (A_1a+A_2, B_1b+B_2)$ will be continuous. (their $F^{-1}$ are, obviously, bijections)

Add:Looks, like if all "ends" of intervals a and b are mapped according to some bijective function, the map is continuous.

Will there be any other?

Edit:replaced relations with maps. Should have looked at the dictionary earlier.

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    By "relations", do you mean **functions**?2011-05-15
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    @Arturo: The definition of preimage can be extended to general relations, and so it can be continuous if the preimage of an open set under the relation is open.2011-05-15
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    notice that any set $(a,b)$ can be written as $\bigcup_{n\in\mathbb N}(a,b-\frac{1}{n}]$, so $T'$ refines $T$ (that is $T\subseteq T'$, since any open set in $T$ can be written as a union of basic open sets of $T'$). Let $f:T\to T$ be a continuous function (in the open ball topology), and let $U\subseteq T$ be open, so $f^{-1}(U)$ is open in $T$, and hence in $T'$. So we know that any continuous function from $T$ to $T$, is also continuous as a $T'\to T$ function. The same arguments works for relations as well. I'm not sure about why these are the only continuous ones, though i think they are.2011-05-15
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    @kneidell: they aren't. For example, the indicator function of $(0,1]$ is $(T',T)$-continuous but not $(T,T)$-continuous. In fact, a function is $(T',T)$ continuous iff it is *left-continuous* in the usual sense.2011-05-15
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    @ChrisEagle Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2015-04-28

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