Topology and its Applications最新文献

A continuum is a nondegenerate, compact, connected, metric space. A topological property P is invariant under retraction provided that each retract of a continuum having P has P, and P is reversible under retraction by pseudo-deformation if the condition a subcontinuum of a continuum X has P implies that X has P. In this paper, we prove that the absence of $R^3$-sets, the absence of $R^4$-continua and the absence of s-points are reversible under retractions by pseudo-deformation, and the absence of $R^3$-sets and the absence of $R^4$-continua are invariant under retractions while the absence of s-points is not.

We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in [1]. In particular, following the theory of quantum invariants we work with ‘rotational’ virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.

In these notes we introduce and investigate two new games called R-nw-selective game and the M-nw-selective game. These games naturally arise from the corresponding selection principles involving networks introduced in [5].

For an inverse sequence on $[0,1]$ with interval-valued functions, we establish necessary conditions on the bonding functions for chainability of the inverse limit space. We also characterize chainability of the inverse limit in this setting in terms of properties of the bonding functions $f_i$ and the induced functions $F_n:[0,1]\to G^{\prime }(f_1,\dots ,f_{n-1})$. The properties, in both cases, are related to how triods arise in the partial graphs associated with the inverse sequence when each graph $G\left(f_i\right)$ is chainable.

In the paper, we study a digital topological complexity of a digital map and its properties. Firstly, we discuss a strong digital homotopy which allows iterative algorithms based on our previous work. As a generalization of the topological complexity in terms of the strong digital homotopy (we call it strong digital topological complexity), we next study the strong digital f-sectional category of a strong digital fibration. Then we investigate estimates of the upper and lower bounds for the strong digital topological complexity of digital maps. We also reveal the difference between the strong digital topological complexity and the ordinary digital ones. It has shown that the strong digital topological complexity is more similar to the classical continuous case than the ordinary digital ones. Moreover, arising from practical considerations in robotics, we consider the naive digital topological complexity of digital maps.

An ultrametric preserving function f is said to be strongly ultrametric preserving if ultrametrics d and $f\circ d$ define the same topology on X for each ultrametric space $(X,d)$. The set of all strongly ultrametric preserving functions is characterized by several distinctive features. In particular, it is shown that an ultrametric preserving f belongs to this set iff f preserves the property to be compact.

Let $f:X\to Y$ be a continuous surjection of compact Hausdorff spaces. By$f_{⁎}:M\left(X\right)\to M\left(Y\right),\mu \mapsto \mu \circ f^{-1}\text{ and }{2}^f:2^X\to 2^Y,A\mapsto f\left[A\right]$ we denote the induced continuous surjections on the probability measure spaces and hyperspaces, respectively. In this paper we mainly show the following facts:

• (1)

  If $f_{⁎}$ is semi-open, then f is semi-open.

• (2)

  If f is semi-open densely open, then $f_{⁎}$ is semi-open densely open.

• (3)

  f is open iff $2^f$ is open.

• (4)

  f is semi-open iff $2^f$ is semi-open.

• (5)

  f is irreducible iff $2^f$ is irreducible.

In this note, it is proved that (1) if $(G,\tau ,\oplus )$ is a strongly topological gyrogroup and H is a closed strong subgyrogroup of G, then $G/H$ is κ-Fréchet-Urysohn if and only if $G/H$ is strongly κ-Fréchet-Urysohn under the condition that H is neutral; (2) let H be a closed strong subgyrogroup of a strongly topological gyrogroup$(G,\tau ,\oplus )$, then the equality $\Delta (G/H)=\psi (G/H)$ holds when H is neutral; (3) if $(G,\tau ,\oplus )$ is a sequential strongly topological gyrogroup having a point-countable k-network, then G is metrizable or a topological sum of cosmic subspaces. There results improve the related results in topological groups.

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product $X=\prod X_i$ of almost discrete spaces $X_i$ the space $C_p\left(X\right)$ of all continuous real-valued functions with the topology of pointwise convergence is a μ-space if, and only if, X is a weak q-space if, and only if, $t\left(X\right)=\omega$ if, and only if, X is functionally generated by the family of all its countable subspaces.

This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC, obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that $X\times Y$ is not weakly Grothendieck space. In $\left(PFA\right)$: the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.

We introduce the selection principles H-star-Lindelöf, weakly H-star-Lindelöf and almost H-star-Lindelöf and we provide some propositions and relationships with other known properties in literature. Furthermore, in this paper we characterize the above selection principles in hyperspaces endowed with the hit-and-miss topology ${\tau }_{\Delta }$, by using a new property ${\mathrm{SS}}_{{\Pi }_{\Delta }\left(\Lambda \right),\text{fin}}^{\star }\left(C_{\Delta }\right(\Lambda ),B)$, for the corresponding choose of the family $B$.