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Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrapower. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions. Furthermore, the two above results can be generalized to larger cardinals with the same proof. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. To prove these facts, one uses a generalization of the notion of separability called the density and the generalized proof is very similar to the original proof. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I hope the above results clear up any confusion about the importance of separability in non-metrizable spaces.

Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrapower. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions (the proofs using independent sets and independent partitions are essentially the same proof. See Andreas Blass's comment below). Furthermore, the two above results can be generalized to larger cardinals with the same proofs. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. To prove these facts, one uses a generalization of the notion of separability called the density and the generalized proof is very similar to the original proof. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I hope the above results clear up any confusion about the importance of separability in non-metrizable spaces.

Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrafilter. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions. Furthermore, the two above results can be generalized to larger cardinals with the same proof. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. To prove these facts, one uses a generalization of the notion of separability called the density and the generalized proof is very similar to the original proof. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I hope the above results clear up any confusion about the importance of separability in non-metrizable spaces.

Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrapower. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions. Furthermore, the two above results can be generalized to larger cardinals with the same proof. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. To prove these facts, one uses a generalization of the notion of separability called the density and the generalized proof is very similar to the original proof. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I hope the above results clear up any confusion about the importance of separability in non-metrizable spaces.

Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrafilter. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions. Furthermore, the two above results can be generalized to larger cardinals with the same proof. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I will probably add some to this post later.

Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrafilter. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions. Furthermore, the two above results can be generalized to larger cardinals with the same proof. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. To prove these facts, one uses a generalization of the notion of separability called the density and the generalized proof is very similar to the original proof. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I hope the above results clear up any confusion about the importance of separability in non-metrizable spaces.