If X is a dense subspace of a space B, then B is called an extension of X, and the subspace Y = $B\setminus X$ is called a remainder of X. We study below, how the properties of remainders of spaces influence the properties of these spaces. In particular, we establish the following fact: if Y is a remainder of a topological group G in an extension B of G, and every closed pseudocompact $G_{\delta }$-subspace of Y is compact, and B contains a nonempty compact subset Φ of countable character in B such that $G\cap \mathrm{\Phi }\ne \varnothing$, then G is a paracompact p-space (Theorem 2.3). This fact plays a key role in the proofs of the similar statements for images and preimages of topological groups under perfect mappings (see Theorems 3.1, 3.2 and 3.4).

如果X是一个空间的子空间密集乙，然后乙被称为扩展的X，并且子空间ÿ =$乙\setminus X$被称为剩余的X。我们在下面研究空间余数的性质如何影响这些空间的性质。特别是，我们建立以下事实：如果ý是拓扑组的剩余部分ģ在延伸乙的ģ，每个闭合pseudocompact$G_{\delta }$的-subspace ÿ紧凑，乙包含可数字符的非空紧子集Φ在乙使得$G\cap \mathrm{\Phi }\ne \varnothing$，则G是超紧致p-空间（定理2.3）。在完全映射下，关于拓扑组的图像和原像的类似陈述的证明中，这一事实起着关键作用（请参见定理3.1、3.2和3.4）。