An ideal on \(\mathbb N\) is a family of subsets of \(\mathbb N\) closed under the operations of taking finite unions and subsets of its elements. The \(\mathcal {I}\)-open sets of topological spaces, which are determined by an ideal \(\mathcal {I}\) on \(\mathbb N\) and the topology of the spaces, are a basic concept of ideal topological spaces. However, it encounters some difficulties in the study of certain structures and mappings of topological spaces. In this paper, we discuss some properties of ideal topological spaces based on \(\mathcal {I}_{sn}\)-open sets, study the problem generating new topological spaces from ideals, characterize the mappings preserving \(\mathcal {I}\)-convergence and structure special \(\mathcal {I}\)-quotient spaces. The following main results are obtained.

1. (i)

   A mapping \(f:X\rightarrow Y\) preserves \(\mathcal {I}\)-convergence if and only if provided U is an \(\mathcal {I}_{sn}\)-open subset of Y, then \(f^{-1}(U)\) is an \(\mathcal {I}_{sn}\)-open subset of X.

2. (ii)

   A topological space X is an \(\mathcal {I}\)-neighborhood space if and only if every \(\mathcal {I}\)-continuous mapping on the space X preserves \(\mathcal {I}\)-convergence.

3. (iii)

   Suppose that both X, Y are topological spaces and \(f:X\rightarrow Y\) is a surjective mapping. Then the topology \(\mu \) of the space Y is the finest topology that makes f preserve \(\mathcal {I}\)-convergence if and only if \(\mu =\tau _{f, \mathcal {I}_{sn}}\), if and only if f is an \(\mathcal {I}_{sn}\)-quotient mapping and \(\mu =\mu _{\mathcal {I}_{sn}}\).

4. (iv)

   Let X be an \(\mathcal {I}\)-neighborhood space and \(f:X\rightarrow Y\) be a surjective mapping. Then the topology \(\mu \) of the space Y is the finest topology that makes f be \(\mathcal {I}\)-continuous if and only if \(\mu =\tau _{f, \mathcal {I}}\), if and only if f is an \(\mathcal {I}\)-quotient mapping and Y is an \(\mathcal {I}\)-sequential space.

These show the unique role of \(\mathcal {I}\)-neighborhood spaces in the study of ideal topological spaces and present a version using the notion of ideals.

上的理想\（\ mathbbÑ\）是子集的一个家族\（\ mathbbÑ\）下取有限工会和它的元素的子集的操作关闭。的\（\ mathcal {I} \）拓扑空间，这是由一个理想的确定的-open套\（\ mathcal {I} \）上\（\ mathbbÑ\）和空间的拓扑结构，是一个基本理想拓扑空间的概念。但是，它在研究拓扑空间的某些结构和映射时遇到一些困难。在本文中，我们讨论基于\（\ mathcal {I} _ {sn} \） -开放集的理想拓扑空间的一些性质，研究从理想中生成新拓扑空间的问题，表征保留的映射\（\ mathcal {I} \）-收敛并构造特殊\（\ mathcal {I} \）-商空间。得到以下主要结果。

1. （一世）

   映射\（F：X \ RIGHTARROW Y \）蜜饯\（\ mathcal {I} \）当且仅当提供-convergence Ú是\（\ mathcal {I} _ {SN} \） -open的子集ÿ，然后\（F ^ { - 1}（U）\）是\（\ mathcal {I} _ {SN} \） -open子集的X。

2. （ii）

   拓扑空间X是\（\ mathcal {I} \） -neighborhood空间当且仅当每\（\ mathcal {I} \） -连续的空间映射X蜜饯\（\ mathcal {I} \） -收敛。

3. （iii）

   假设X，  Y都是拓扑空间，\（f：X \ rightarrow Y \）是一个射影映射。然后，空间Y的拓扑\（\ mu \）是使f保留\（\ mathcal {I} \）-收敛的最佳拓扑，当且仅当\（\ mu = \ tau _ {f，\ mathcal { I} _ {sn}} \），并且仅当f是\（\ mathcal {I} _ {sn} \）-商映射和\（\ mu = \ mu _ {\ mathcal {I} _ { sn}} \）。

4. （iv）

   令X为\（\ mathcal {I} \）-邻域空间，而\（f：X \ rightarrow Y \）为射影映射。那么，空间Y的拓扑\（\ mu \）是使f成为\（\ mathcal {I} \） -连续且当且仅当\（\ mu = \ tau _ {f，\ mathcal { I}} \），并且仅当f是\（\ mathcal {I} \）-商映射且Y是\（\ mathcal {I} \）-序列空间。

这些显示了\（\ mathcal {I} \）-邻域空间在理想拓扑空间研究中的独特作用，并提出了使用理想概念的版本。