一致空间范畴中直接极限的拓扑结构
The topological structure of direct limits in the category of uniform spaces
摘要：
设$(X_n)_{n}$是一个一致空间序列，使得每个空间$X_n$是$X_{n+1}$中的一个闭子空间。在一致空间范畴中，给出了序列$(X_n)$的直接极限$u-lim X_n$的拓扑和一致性的一个明确描述。这个描述意味着函数$f:u-lim X_n\to Y$到均匀空间$Y$是连续的，如果对于每个$n$，限制$f|X_n$在子集$X_{n-1}$处是连续的和正则的，在这个意义上，对于任何随行$U\in\U_Y$和$V\in\U_X$，存在随行$V\in\U_X$，使得对于每个点$x\in B(X_{n-1},V)$，存在具有$(x,x')\in V$和$(f(x),f(x'))\in U$的点$x'\in X_{n-1}$。此外，我们将比较拓扑结构的直接限制在各种类别。
英文摘要：
Let $(X_n)_{n}$ be a sequence of uniform spaces such that each space $X_n$ is a closed subspace in $X_{n+1}$. We give an explicit description of the topology and uniformity of the direct limit $u-lim X_n$ of the sequence $(X_n)$ in the category of uniform spaces. This description implies that a function $f:u-lim X_n\to Y$ to a uniform space $Y$ is continuous if for every $n$ the restriction $f|X_n$ is continuous and regular at the subset $X_{n-1}$ in the sense that for any entourages $U\in\U_Y$ and $V\in\U_X$ there is an entourage $V\in\U_X$ such that for each point $x\in B(X_{n-1},V)$ there is a point $x'\in X_{n-1}$ with $(x,x')\in V$ and $(f(x),f(x'))\in U$. Also we shall compare topologies of direct limits in various categories.
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