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Suppose $(X,\tau_X)$ is a topological space, and $Y \subset X$ a subset. We can equip $Y$ with the topology $\tau_Y:= \{Y \cap A: A\in \tau_X \}$, called the subspace topology.

When one says "$W \subset Y$ is open", does it mean $W\in \tau_X$ and $W\subset Y$ or does it mean $W\in \tau_Y$?

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    $\begingroup$ When one says that, one can avoid confusion by saying "... is open in $Y$" or "... is open in $X$". $\endgroup$ – Robert Israel Jun 16 '16 at 17:19
  • $\begingroup$ It happens that "$W \subset Y$ is open" can mean both (which brings us back to @RobertIsrael's comment). Rather surprised by the definitive tone of the answers below. $\endgroup$ – Did Jun 20 '16 at 10:29
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It means the latter: one can have sets which are open in the subspace topology, but aren't in the topology of the bigger space.

Example: Consider the normal topology on $\mathbb{R}$, and consider the subspace topology on $[0,1]$. Clearly, $(0,1]$ is open in $[0,1]$, since $(0,1] = (0,2) \cap [0,1]$, but $(0,1]$ is not open in $\mathbb{R}$.

To use your notation, $W = (0,1]$ does not belong to $\tau_X$, hence does not satisfy the first definition.

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It means $W=V\cap Y$ where $V\in \tau_X$.

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    $\begingroup$ which also means $W\in \tau_Y$, with the subspace topology. $\endgroup$ – snulty Jun 16 '16 at 17:06
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It means the latter. The subspace topology is a surprisingly slippery concept for beginning students of topology to grasp. Like linear subspaces of vector spaces in linear algebra, a topological subspace $A\subseteq X$ of a topological space X is itself a topological space whose topology is induced by the larger topology,but is a different unique topology on A.

Here's a simple example. Consider the real line with the usual metric topology. Consider the interval (-$\infty$,1] as a subspace of the real line. Now consider the interval (0,1]. Is this interval an open subspace in $\mathbb R$ ? Under the usual topology, no. But is it relatively open in the subspace (-$\infty$,1]? YES, because (0,1]= (-$\infty$,1]$\cap$ (0,2) and by definition, this lies in the subspace topology on (-$\infty$,1]!

The best way to understand the subspace topology is to study and construct a lot of examples until you get really comfortable with the concept.

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