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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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    @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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