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In connection with the question continuous onto map from $(0,1)\to (0,1]$ I would like to know whether $(0,1)$ and $(0,1]$ are homeomorphic. The map mentioned in the above question is onto but not a bijection. So does such a continuous bijection exist?

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Sriti Mallick
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HINT: For every $x\in(0,1)$, $(0,1)\setminus\{x\}$ is not connected. If $(0,1)$ and $(0,1]$ were homeomorphic, the same would be true of $(0,1]$, since connectedness is a topological property. Does $(0,1]$ actually have this property, or does it have some point that can be removed without disconnecting it?

Brian M. Scott
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  • @Sriti: That deleting any point leaves a space that is not connected. In other words, the question is: Is it true that $(0,1]\setminus{x}$ is disconnected for every $x\in(0,1]$? – Brian M. Scott Aug 02 '13 at 10:40