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Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase

$$\Lambda =\{ \{(f,g)\in X^X\times X^X\mid (\forall a\in A) \big((f(a),g(a))\in E\big)\} \mid A\in \mathcal A, D\in \mathcal D\}$$ where $\mathcal A$ is the set of all closed sets.

Let $m,n:(D,\le)\to G$ be two nets converging to $f,g\in G$ respectively. How can I prove $$n_x\circ m_x \to f\circ g$$ ?

Something that may solve my problem is how to describe a normal uniform space directly. Every uniform space is completely regular but how can a normal uniform space be described directly by adding some property to entourages?

Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase

$$\Lambda =\{ \{(f,g)\in X^X\times X^X\mid (\forall a\in A) \big((f(a),g(a))\in E\big)\} \mid A\in \mathcal A, D\in \mathcal D\}$$ where $\mathcal A$ is the set of all closed sets.

Let $m,n:(D,\le)\to G$ be two nets converging to $f,g\in G$ respectively. How can I prove $$n_x\circ m_x \to f\circ g$$ ?

Something that may solve my problem is how to describe a normal uniform space directly. Every uniform space is completely regular but how can a normal uniform space be described directly by adding some property to entourages?

Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase

$$\Lambda =\{ \{(f,g)\in X^X\times X^X\mid (\forall a\in A) \big((f(a),g(a))\in E\big)\} \mid A\in \mathcal A, D\in \mathcal D\}$$ where $\mathcal A$ is the set of all closed sets.

Let $m,n:(D,\le)\to G$ be two nets converging to $f,g\in G$ respectively. How can I prove $$n_x\circ m_x \to f\circ g$$ ?

Something that may solve my problem is how to describe a normal uniform space directly. Every uniform space is completely regular but how can a normal uniform space be described directly by adding some property to entourages?

Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase

$$\Lambda =\{ \{(f,g)\in X^X\times X^X\mid (\forall a\in A) \big((f(a),g(a))\in E\big)\} \mid A\in \mathcal A, D\in \mathcal D\}$$ where $\mathcal A$ is the set of all closed sets.

Let $m,n:(D,\le)\to G$ be two nets converging to $f,g\in G$ respectively. How can I prove $$n_x\circ m_x \to f\circ g$$ ?

Something that may solve my problem is how to describe a normal uniform space directly. Every uniform space is completely regular but how can a normal uniform space be described directly by adding some property to entourages?