Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\mathscr{S}$ is the unit ball of $\mathscr{M}$.

Why exactly can we approximate a point in the strong closure of $E^*p\cap r\mathscr{S}$ by a sequence (as opposed to a net)? I imagine it has something to do with metrizability of $p\mathscr{M}p$? Another explanation is that $\mathscr{M}_p$ is a typo and it should be $\mathscr{M}p \cap \mathscr{S}$ that should be metrizable? But in that case I don't see why this must be metrizable.

Any help/clarification is highly appreciated!

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\mathscr{S}$ is the unit ball of $\mathscr{M}$.

Why exactly can we approximate a point in the strong closure of $E^*p\cap r\mathscr{S}$ by a sequence (as opposed to a net)? I imagine it has something to do with metrizability of $p\mathscr{M}p$? Another explanation is that $\mathscr{M}_p$ is a typo and it should be $\mathscr{M}p \cap \mathscr{S}$ that should be metrizable? But in that case I don't see why this must be metrizable.

Any help/clarification in the matter is highly appreciated!

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\mathscr{S}$ is the unit ball of $\mathscr{M}$.

Why exactly can we approximate a point in the strong closure of $E^*p\cap r\mathscr{S}$ by a sequence (as opposed to a net)? I imagine it has something to do with metrizability of $p\mathscr{M}p$? Another explanation is that $\mathscr{M}_p$ is a typo and it should be $\mathscr{M}p \cap \mathscr{S}$ that should be metrizable? But in that case I don't see why this must be metrizable.

Any help is highly appreciated!

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\mathscr{S}$ is the unit ball of $\mathscr{M}$.

Why exactly can we approximate a point in the strong closure of $E^*p\cap r\mathscr{S}$ by a sequence (as opposed to a net)? I imagine it has something to do with metrizability of $p\mathscr{M}p$? Another explanation is that $\mathscr{M}_p$ is a typo and it should be $\mathscr{M}p \cap \mathscr{S}$ that should be metrizable? But in that case I don't see why this must be metrizable.

Any help/clarification is highly appreciated!

Consider the following fragments from Takesaki's second voluem "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\mathscr{S}$ is the unit ball of $\mathscr{M}$.

Why exactly can we approximate a point in the strong closure of $E^*p\cap r\mathscr{S}$ by a sequence (as opposed to a net)? I imagine it has something to do with metrizability of $p\mathscr{M}p$? Another explanation is that $\mathscr{M}_p$ is a typo and it should be $\mathscr{M}p \cap \mathscr{S}$ that should be metrizable? But in that case I don't see why this must be metrizable.

Any help is highly appreciated!

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\mathscr{S}$ is the unit ball of $\mathscr{M}$.

Why exactly can we approximate a point in the strong closure of $E^*p\cap r\mathscr{S}$ by a sequence (as opposed to a net)? I imagine it has something to do with metrizability of $p\mathscr{M}p$? Another explanation is that $\mathscr{M}_p$ is a typo and it should be $\mathscr{M}p \cap \mathscr{S}$ that should be metrizable? But in that case I don't see why this must be metrizable.

Any help is highly appreciated!