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We know $(0,1)$ is open in $\mathbb{R}$. Please explain if $(0,1)$ is open in $(0,1]$ or not.

How to do that?

Topology
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    use definitions: what is an open set in the (usual) metric topology of $\mathbb R$? Can you apply the definition to $(0,1]$, if you include $1$ on the right? – Avitus Dec 16 '13 at 12:36
  • Subspace topology is defined as an intersection of the given subset with all open sets in the original set. Since $\left( {0,1} \right)$ is open in $\mathbb{R}$, the answer becomes obvious – Alen Dec 16 '13 at 12:37

3 Answers3

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When you have $X$ a subspace of a topological space $Y$, the open sets of $X$ are by definition the sets of the form $U \cap X$, where $U$ is open in $Y$. Take $X = [0,1]$ and $Y = \mathbb{R}$. You should conclude easily.

Jeremy Daniel
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Use subspace topology. $$(0,1)=(0,1)\cap(0,1]$$

user113578
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Since this is tagged as metric-spaces, I'll assume you don't know about topology.

Remember, a set $A$ in a metric space is open if, for all $x \in A$, there's some open ball $x \in \{y : d(x,y) < r\} \subset A$. So pick $x \in (0,1)$, and then find a ball small enough around $x$ that's contained in $(0,1)$...