1)  topological space

1.
Seven definitions of topological space and their sameness;

2.
Some Browder type fixed point theorems in topological spaces with applications;

3.
Some properties of the relative topological space;

2)  topological spaces

1.
Fractal dimensions of polyferric chloride-humic acid(PFC-HA) flocs in different topological spaces;

2.
A semigroup of closed selfmaps of a kind of topological spaces;

3.
The physical properties and fractal dimensions within different topological spaces of the mature granular sludge in an anaerobic baffled reactor (ABR) were investigated.

3)  topology space

1.
In reference 1 ,the theorems about fibre boundness and compactness of uniform space with shadow to be topology space were given.

2.
A cover U of a topology space X has a alternate σ-relatively locally finite and relatively closed refinement.

4)  Spatial Topology

1.
From the point of the cartography history,according to some ancient Chinese maps and foreign maps,three constants of sign,lettering and spatial topology relationship were put forward along with the map progress,which consisted of the basic elements about cognizing the map spatial relationship.

5)  quotient topological space

1.
Describe the logical relations among mining survey objects using quotient topological space;

6)  type topological space

1.
In F type topological space which introduces the auxilliary order,the fixed point theorems of increasing mappings at order intervals are proved based on the relative properties

 拓扑空间topological space   赋予拓扑结构的集合。如果对一个非空集合X给予适当的结构，使之能引入微积分中的极限和连续的概念，这样的结构就称为拓扑，具有拓扑结构的空间称为拓扑空间。引入拓扑结构的方法有多种，如邻域系、开集系、闭集系、闭包系、内部系等不同方法。下面介绍开集系方法。在微积分学中，实一维欧几里得空间R′上的开集具有性质：①任意个开集的并是开集 。②有限个开集的交是开集。③R′及空集是开集。对任一非空集合X，若X的一个子集族J满足：①J中元的任意并在J中。②J中元的有限交在J中。③X、在J中，则称J是X的一个拓扑，J中的元称为开集，X连同拓扑J称为一个拓扑空间，记为（X，J）。   对任意x∈X，如果Z的子集U包含含有x的一个开集则U称为x的一个邻域。如果X的子集A满足X－A是开集，则称X是闭集。   设X是非空集合，令J0＝｛X，｝，称（X，J0）为平庸拓扑空间，J0为平庸拓扑。令J1＝｛A｜AÌX｝，称（X，J1）为离散拓扑空间。在离散拓扑空间中任意子集均是开集。对实数集R1，令J＝｛BÌR1｜"x∈G，∈ε＞0，使（x－ε，x＋ε）ÌG｝，则（R1，J）就是一维欧几里得空间。类似地可定义n维欧几里得空间Rn。    设X是拓扑空间，如果X可写为非空开集的分离并，则X称为连通空间；如果对X中任意两点  ，存在X中的道路相连接，则称X为道路连通空间  ；如果X的任意开集作成的覆盖存在有限子覆盖  ，则称X为紧空间；如果X中的任意序列有收敛子列，则称X是列紧空间  ；如果X中任意两点都存在不相交的邻域  ，则称X是豪斯多夫空间（或T2空间）。上面所提连通性，道路连通性、紧性、列紧性、T2性均是拓扑不变性。连通空间上的实值连续函数具有介值性，即若f∶X→R1连续，X是连通空间，r∈（f（x1），f（x2），则存在c∈（x1，x2）（或c∈（x2，x1）），使f（c）＝r。紧空间上的实值连续函数具有最大值、最小值。紧空间上的连续函数一致连续。若AÌRn，则A为紧，当且仅当A是有界闭集。