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I am currently reading an introduction to topological and metric spaces and want to know whether the following statement is true:

Consider the Euclidean space $\mathbb{R}^n$ endowed with the Euclidean metric. Any function that maps an open ball in $\mathbb{R}^n$ to another open ball is homeomorphic.

It is clear to me that any function $f:X\rightarrow T$, with $X$ the discrete topology and T an arbitrary topology, is homeomorphic. Is the beforementioned statement somehow linked to this?

Thx

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    Um: what do you mean by "homeomorphic"? The word "homeomorphic" is an _adjective_ describing a relationship between two objects. So it doesn't make sense to say that "[A] function ... is homeomorphic." A function between two objects may be a "homeomorphism", if through it we can see that the two objects are homeomorphic. But given the third paragraph of your question, perhaps the adjective you are looking for is "[continuous](http://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces)"?2011-09-12
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    It is not true that every map $f:X\rightarrow T $ with $X$ discrete is a homeomorphism; for one thing, the discrete topology is metrizable, so Hausdorff, and Hausdorff is a topological property, so if $T$ has any non-Hausdorff topology, then $f:X\rightarrow T$ cannot be a homeomorphism.2011-09-12
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    People seem to be going to an awful lot of trouble to describe situations where $f:X\rightarrow T$ cannot be a homeomorphism when $X$ is discrete... isn't "T does not have the discrete topology" sufficient?2011-09-12
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    Well, I gave specific examples because I like to see specific examples in areas I'm not familiar with, and I imagined others in similar conditions would too.2011-09-12
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    I am speculating here, but based on the OP's last paragraph I think he is confusing the terms "continuous function" and "homeomorphism".2011-09-12

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