Michal Stanislaw Wojcik
The main result of this paper states that for a function $f:\R^2\to Y$ with a closed, connected and locally connected graph, where $Y$ is a locally compact, second-countable metrisable space, the graph over discontinuity points remains locally connected.
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The main result of this paper states, that if a function $f:\R^2\to [0, +\infty)$ has a closed graph and the set of discontinuity points is a network (as defined by Kuratowski in Topology II, 61.IV), then the graph of $f$ is disconnected. It is also proven that this result can be easily generalised to a function $f:\R^2\to Y$ where $Y$ is a $σ$-locally compact metrisable space.
Rings of functions whose closure of discontinuity set is in an ideal of closed sets
本文研究不连续点集闭包属于给定闭集理想的实值函数环的环性质，并给出反例说明非 Tychonoff 空间上“每个 P-空间都是 FP-空间”的结论可能不成立。
In an exploration paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, we designed algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. However, the algorithms could not guarantee to be applicable to every case. This paper will be the continuation of the exploration. This paper contains two main procedures: (1) A shrinking procedure to contract a simply connected closed manifold. Unlike ones in the previous paper, we added a tree structure to support the process. (2) A more direct pr…
In this exploration paper, we design algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. Such a contraction is a kind of shrinking or reducing process. In our algorithms, we need to assume an ambient space for the discrete manifold, and this ambient space also a simply connected discrete space in higher dimensions. Our algorithm would work for most of cases. For some special cases, we will make detailed analysis in the next paper. In other words, this paper has not provided a complete proof for each case. The algorithm designed in this pap…
A topological group $X$ is called connected if the only subsets which are both open and closed are the whole space $X$ and the null set $\emptyset$. A subset of a topological group is connected if the subspace is connected. We say that a subset $A$ of $X$ is $G$-sequentially connected if the only subsets of $A$ which are both $G$-sequentially open and $G$-sequentially closed, with respect to the relative $G$-sequentially open and $G$-sequentially closed subsets of $A$, are open and closed subsets of $A$ are $A$ and the null set, $\emptyset$. We investigate the impact of changing the definition…
We prove that each closed locally continuum- connected subspace of a finite dimensional topological group is locally compact. This allows us to construct many 1-dimensional metrizable separable spaces that are not homeomorphic to closed subsets of finite-dimensional topological groups, which answers in negative a question of D.Shakhmatov. Another corollary is a characterization of Lie groups as finite-dimensional locally continuum-connected topological groups. For locally path connected topological groups this characterization was proved by Gleason and Palais in 1957.
A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, locally compact abelian groups and connected locally compact groups and is closed under products and closed subgroups. A topological group G is almost connected if the quotient group of G by the connected component of its identity is compact…
Connected sums of closed Riemannian manifolds and fourth order conformal invariants
本文对共形几何中Yamabe问题的一个四阶类似问题作了初步的研究。所考虑的Paneitz常数和Paneitz不变量被认为对理解带下划线流形的拓扑是非常有帮助的。我们计算这些量如何变化，类似于Yamabe常数和Yamabe不变量在连接和操作下的变化。
In this paper, it is proved that every topological gyrogroup $G$ is topologically groupoid isomorphic to a closed subgyrogroup of a connected, locally connected topological gyrogroup $G^{\bullet}$.
The characteristic index of a locally compact connected group $G$ is the non-negative integer $d$ for which we have a homeomorphism $G\cong K\times \mathbb{R}^d$ with $K\le G$ maximal compact. We prove that the characteristic indices of closed connected subgroups are dominated by those of the ambient groups.
Every proper closed subgroup of a connected Hausdorff group must have index at least c, the cardinality of the continuum. 70 years ago Markov conjectured that a group G can be equipped with a connected Hausdorff group topology provided that every subgroup of G which is closed in all Hausdorff group topologies on G has index at least c. Counter-examples in the non-abelian case were provided 25 years ago by Pestov and Remus, yet the problem whether Markov's Conjecture holds for abelian groups G remained open. We resolve this problem in the positive.
Closed graph property in Alexandroff spaces
本文证明在Alexandroff空间中，映射具有闭图当且仅当在其每个连通分支上具有常闭值，且具有闭图意味着连续，并指出此类映射数量仅取决于连通分支和闭点数量。
In this note we give an example of a one-dimensional manifold with two connected components and a complete metric whose group of isometries has an orbit which is not closed. This answers a question of S. Gao and A. S. Kechris.
On scatteredly continuous maps between topological spaces
拓扑空间之间的映射$f:X\to Y$被定义为{ \em散射连续}，如果对于每个子空间$A\subset X$，约束$f|A$有连续点.证明了对于从完全仿紧的遗传Baire Preiss-Simon空间$X$到正则空间$Y$的函数$f:X\to Y$，$f$的离散连续等价于（i）弱间断（对于每个子集$A\subset X$，$f|A$的不连续点的集合$D(f|A)$在$A$中没有密集的地方），（ii）$σ$ -连续性（$X$可以写为闭子集的可数并集，在闭子集上$f$是连续的），（iii）$G_δ$ -可测性（每个开集的原像是$G_δ$类型）。同样在Martin公理下，我们在可度量化可分空间之间构造了一个$G_δ$ -可测映射$f:X\to Y$，它不是分段连续的.这回答了维诺库罗夫的一个老问题。
We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ is compact; (2) $Y$ is a 2-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 2$; (3) $Y$ is a 3-manifold and $X$ is a simply-connected closed $n$-manifold of dimension $n\ge 3$. This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow.
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