Papers

We prove that every chain-continuous dcpo is continuous and that its way-below relation is the relational square of its chain-below relation, answering a question of Erné. A subalgebra lattice obtained from a locally finite Jónsson algebra shows that the chain-below relation itself need not interpolate. For approximation by Scott-closed sets, we give a finite-upper-bound characterization on lower-set lattices and characterize interpolation by an extension property for directed lower sets. This yields a countably based, completely distributive algebraic lattice on which chain and directed approximation coincide, whereas Scott-closed approximation fails to interpolate, answering questions of Erné and of Ho and Zhao.

In this paper, we investigate the family of $H^{*}$-open sets from the perspective of generalized topology and examine the structural and separation properties arising from it. For a topological space $(\mathbb{X},\tau)$, the family of all $H^{*}$-open subsets of $\mathbb{X}$ is considered as a natural candidate for an $H^{*}$-generated generalized topology. We establish conditions under which this family forms a strong generalized topology and identify additional assumptions that allow the resulting structure to be treated as an ordinary topology. This distinction provides a systematic framework for studying $H^{*}$-open sets without imposing finite-intersection properties in advance. Within this framework, we investigate the corresponding $H^{*}$-interior and $H^{*}$-closure operators and analyze separation properties determined by the $H^{*}$-open structure. When the $H^{*}$-generated family forms a topology, the resulting associated space is used to relate generalized $H^{*}$-separation properties to their classical topological counterparts. We further examine compactness-type properties, subspace behaviour, and preservation under suitable mappings. Examples are included to illustrate how the structure generated by $H^{*}$-open sets may exhibit separation behaviour different from that of the original topology. The results provide a unified setting for connecting $H^{*}$-open-set theory with generalized topological structures and offer a basis for further investigation of separation and covering properties in this framework.

In this paper, we focus on an open problem whether every locally compact dcpo is sober for the Scott topology. Specifically, we obtain two sufficient conditions for answering this problem. The main results are (1) every meet continuous locally compact coherent dcpo is sober with respect to the Scott topology; (2) the Scott space of every locally compact second countable dcpo is a sober space.

We consider several families of sequences of Baire-one self-maps of $[0,1]$ investigated by A. Alikhani-Koopaei. They form a chain of inclusions where each space is closed and nowhere dense in the next one. We prove that each space is lower $2/3$-porous in the next one. If one modifies these families where sequences of Baire-one functions are real-valued uniformly bounded, then they are Banach spaces and each space of the respective chain is c-porous in the next one.

Fix a finite alphabet $\Sigma$ having at least two letters, and natural numbers $n,k\geq 1$. We denote by $D_n(\mathbf{\Pi}^0_k)$ the class of differences of $n$ subsets of the Cantor space $\Sigma^\omega$ in $\mathbf{\Pi}^0_k$, and by $\checked{D}_n(\mathbf{\Pi}^0_k)$ the class of complements of sets in $D_n(\mathbf{\Pi}^0_k)$. We prove, for any natural numbers $n,k\geq 1$, the existence of a finite alphabet $\Sigma$ having at least two letters and a finitary language $U_{n,k}$ over $\Sigma$ accepted by a one-counter automaton such that the $\omega$-power $$U_{n,k}^\infty:=\{ w_0w_1\cdots\in\Sigma^\omega\mid\forall i\in\omega ~~w_i\in U_{n,k}\}$$ is $D_{2n-1}(\mathbf{\Pi}^0_k)$-complete. We also prove a conditional result for the dual classes $\checked{D}\_{2n}(\mathbf{\Pi}^0\_k)$: under the assumption that there exists a finitary language $V_{n,3}$ accepted by a one-counter automaton, such that its $\omega$-power $V_{n,3}^\infty$ is $\checked{D}_{2n}(\mathbf{\Pi}^0_3)$-complete, there would also exist, for any integer $k>3$, a finitary language $V_{n,k}$ accepted by a one-counter automaton, such that its $\omega$-power $V_{n,k}^\infty$ is $\checked{D}_{2n}(\mathbf{\Pi}^0_k)$-complete. Finally we add another conditionnal result about Wadge degrees of rational $\mathbf{\Delta}^0_2$ omega powers, which completes the recent results of [FGL26].

An elementary space is a pair $(X, \mathcal A)$, where $\mathcal A$ is an algebra of subsets of $X$. Every elementary space $(X, \mathcal A)$ is naturally a topological space when equipped with the topology $\tau_{\mathcal A}$ generated by $\mathcal A$. Elementary mappings on elementary spaces are defined in a manner analogous to measurable mappings on measurable spaces. In addition, the notion of an extension of an elementary space is introduced. Let $M(X, \mathcal A)$, denote the ring of all real-valued elementary functions on $(X, \mathcal A)$. In this article, we study the ring $M(X, \mathcal A)$ and extensions of an elementary space. We show that, for any elementary space $(X, \mathcal A)$, there exists a unique compact elementary extension $(\beta_{\mathcal A}X, \bar{\mathcal A})$ of $X$, analogous to the Stone-$\check C$ech compactification, which contains every other extension of $X$. This extension has the property that every elementary map from $X$ into any compact topological space extends continuously to $\beta_{\mathcal A}X$. Next, we show that $\beta_{\mathcal A}X$ contains a realcompact extension $(\upsilon_{\mathcal A}X, \bar{\mathcal A}_R)$ of $X$, such that every elementary map from $X$ into any realcompact topological space extends continuously to $\upsilon_{\mathcal A}X$. Moreover, $\upsilon_{\mathcal A}X$ is the smallest realcompact extension of $X$ and $(X, \mathcal A)$ is realcompact if and only if $X=\upsilon_{\mathcal A}X$. Also, we see that, every metric space equipped with the $\sigma$-algebra of all Borel sets is realcompact. Finally, we observe that the maximal ring of quotients of $M(X, \mathcal A)$ is the direct limit of locally constant (continuous) functions on dense open subsets of $(X, \tau_{\mathcal A})$. Some questions raised in the literature are also answered.

We construct a Hausdorff quasitopological abelian group that is second-countable and semiregular, with a bounded continuous function that does not factor through any regular first-countable semitopological group. Its regular and Tychonoff reflections coincide, have a countable network, and fail even factorizability through arbitrary first-countable semitopological groups. This gives negative answers to \cite[Problems~6.1, 6.2, 6.4, and~6.5]{Tk2026} of Tkachenko. The construction starts with a countable cone group whose common separation reflection is not Fréchet–Urysohn, answering \cite[Problems~6.8 and~6.9]{Tk2026} negatively. In contrast, symmetrization gives affirmative answers to \cite[Problems~6.6 and~6.7]{Tk2026} without separation assumptions. We also show that every countable semitopological group admits simultaneous factorizations of countable families through second-countable Tychonoff semitopological groups.

I introduce a new class of cell complexes, called CL-complexes, that provide alternative topological realizations of finite posets. Each realization of a given poset has the same homotopy type as the order complex of that, so in some sense these spaces do not provide anything really new. Their value is that they enable short transparent proofs of standard results in poset homotopy. These include Quillen's Theorem A for finite posets, the Nerve Theorem for good covers of a topological space, and Sheehy's multicover generalization of the Nerve Theorem. Order complexes retain a special status: the order complex of a finite poset admits CL-complex structures both for the poset itself and for its opposite poset. The definition of CL-complexes is motivated by an elementary generalization of the Carrier Lemma of J. W. Walker, which in its original form constructs a canonical homotopy class of maps from a simplicial complex to a space covered by a suitable family of contractible sets. In the generalized lemma, the role played by simplicial complexes and the role played by spaces covered by contractible sets can both be fulfilled by CL-complexes. This sets up an equivalence of categories that tacitly underlies our approach.

Graphs defined on groups, such as power graphs, enhanced power graphs and commuting graphs, form a hierarchy under inclusion of edge sets. We develop an analogous theory for topological spaces. To a space X we attach five graphs on the vertex set X, all invariant under homeomorphisms: the specialization, inseparability, closure-meeting, Urysohn and functional graphs. They form a diamond-shaped hierarchy whose members are null exactly when X is T_1, Hausdorff, Urysohn or functionally Hausdorff. The Urysohn graph always contains the square of the inseparability graph, with equality for Alexandrov spaces, and the hierarchy is compatible with products via the strong product of graphs. Every countable graph, together with isolated vertices, is the inseparability graph of a second countable T_1 space. For finite spaces we characterize the graphs that occur as inseparability graphs (every edge lies in the closed neighbourhood of a simplicial vertex), determine all coincidences between members of the hierarchy, and show that the specialization graph determines the weak homotopy type but not the space itself. Using Dowker duality we prove that the complexes of inseparable and of closure-meeting families are always homotopy equivalent and, under an acyclicity hypothesis, recover the weak homotopy type. We show that if the closure-meeting graph is a cograph then it is chordal, an analogue of a recent result for enhanced power graphs; the same holds for the inseparability graph of an Alexandrov space but not in general. Finally, following Cameron's work on topologies preserved by permutation groups, we show that for a topology invariant under a primitive group each graph is null or connected, and that the neighbourhood filters of the five graphs reverse the hierarchy, giving spaces whose specialization graph contains the Rado graph as a spanning subgraph.

We establish a compact-localization theorem for finite iterates of the Smyth power space. Successive union maps assign to every finite-level Smyth object a canonical least compact saturated carrier in the original space, and co-sobriety at each positive finite level is equivalent to co-sobriety of the corresponding iterates over all compact saturated carriers. Consequently, if every compact saturated subspace of a $T_0$-space $X$ is stably compact—in particular, if every such subspace is Hausdorff—then $\PS^n(X)$ is co-sober for every finite $n\geq 1$. This applies to every $k_1$-Hausdorff space and hence to every Hausdorff or metrizable space. For the first iterate, we also give a direct full-Vietoris argument, independent of stable compactness, which shows that each $k$-irreducible compact saturated family is generated by its union.

We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset $A$ of a $T_1$-space $(X,\tau)$, we consider the topology consisting of the co-finite open sets together with the $\tau$-open sets disjoint from $A$. We determine its $k$-irreducible compact subsets and derive a criterion that produces two compact co-sober spaces whose product contains a non-singleton $k$-irreducible compact saturated subset of the diagonal. Applying the criterion to a specific Kovář–van Douwen space gives two countable compact Fréchet–Urysohn $T_1$ anti-Hausdorff co-sober spaces whose product is not co-sober. A further construction yields a countable compact Fréchet–Urysohn $T_1$ anti-Hausdorff co-sober space whose square is not co-sober. The failure of product closure also gives an alternative proof of the known non-reflectivity of co-sober spaces in the category of $T_0$-spaces.

We completely resolve Question~5.3 of Kania's paper \cite{Kania} and establish a more general realization theorem. We show that for infinite cardinals $\mu\leqslant\nu<\kappa<\lambda$, a space with \[ \pc(X)=\mu,\qquad\pcl(X)=\nu,\qquad \pcc(X)=\kappa,\qquad\pcd(X)=\lambda \] exists if and only if $\kappa\leqslant2^\mu$. The witnesses may be zero-dimensional and Tychonoff spaces, of the least possible cardinality $\lambda$. They may have the least possible number of non-isolated points, or alternatively be crowded. The main ingredient is an exact computation of the centered local $\pi$-character of star filter spaces in terms of ultrafilter character, obtained from a finite selection lemma for filter pushforwards. We also construct a single space with countable $r$-linked local $\pi$-bases for every finite $r$, centered value $\omega_1$, and any prescribed decreasing value greater than $\omega_1$.

Let $A(X)$ denote the free Abelian topological group over a Tychonoff space $X$. We prove that if $A(X)$ and $A(Y)$ are topologically isomorphic and $X$ is countably compact, then $Y$ is also countably compact. Thus countable compactness is an $A$-invariant and, consequently, an $M$-invariant; this answers Open Problem~7.10.1 of Arhangel'skii and Tkachenko.

For a Hausdorff locally quasi-convex abelian group $G$, let $\C(G)$ be the poset of all Hausdorff locally quasi-convex group topologies on its underlying group having the same continuous characters as $G$. We prove that, whenever $G$ is non-precompact, $\C(G)$ contains an order-isomorphic copy of $(\Pow(\cont),\subseteq)$, where $\cont=2^{\aleph_0}$. The embedding takes values between the Bohr topology and the original topology. Consequently, every infinite discrete abelian group $D$ satisfies $|\C(D)|=\width\C(D)=2^{2^{|D|}}$. The discrete reduction for locally compact abelian groups then yields exact cardinality and width formulas for all such groups; in particular, both invariants equal $2^{\cont}$ for every non-compact $\sigma$-compact locally compact abelian group. These results answer Questions 6.1–6.3 and 6.5–6.7, and the locally compact case of Question 6.4, posed by L.~Außenhofer and D.~Dikranjan in \cite{AD20}. The embedding also applies to non-compact complete metrizable locally quasi-convex groups. A non-compact precompact nuclear group with a unique compatible topology shows that non-compactness alone does not suffice in the nuclear setting. Finally, the compatible poset of $\R^{\N}$ is not order-isomorphic to that of any discrete abelian group.

The center of distances $C(Y)$ of a metric space $(Y,\rho)$ is the set of all $t\geq 0$ for which the equation $\rho(y,p)=t $ has a solution for each point $p\in Y$. We say that a simple graph $CG_Y$ is the central graph of a metric space $(Y,\rho)$ if $Y$ is the vertex set of $CG_Y$ and distinct vertices $x,y\in Y$ are adjacent if and only if $\rho(x,y)\in C(Y)$. We prove the inequality $|E(CG_Y)| \geq \left\lceil \frac{1}{2}|Y| \right\rceil $ for all finite metric spaces $(Y,\rho)$ with $|Y|\geq 2$ and non-empty $CG_Y.$ It is also proved that the inequality $|E(CG_X)| \geq |X|-1 $ holds for each finite ultrametric space $(X,d)$. The central graphs of metric spaces $(Y,\rho)$ and ultrametric spaces $(X,d)$ satisfying $|E(CG_Y)|=\left\lceil \frac{1}{2}|Y| \right\rceil $ and, respectively, $|E(CG_X)|=|X|-1 $ are described up to graph isomorphism.

In this paper, we first construct a universal quasi-Polish space Y with the property that every quasi-Polish space is homeomorphic to a closed subspace of Y. This gives a hyperspace F(Y) of all quasi-Polish spaces which is itself a quasi-Polish space. We then prove that the collection of all Polish closed subspaces of Y forms a coanalytic, non-Borel subset of F(Y). In particular, this collection is {\Pi}11-complete.

We introduce a systematic way to use the subobject functor on any category to set up a framework for generalized topological and measure-theoretic notions and inferences. Concretely, we develop two connected calculi: 1. a fourfold calculus derived from notions of smallness and density that generalizes, among other things, topological density and measure-theoretic negligibility, 2. a twofold calculus that generalizes topological and measure-theoretic inference procedures. The resulting theory is distinguished in that it is inter-categorical in character: its definitions can be applied not just to describe smallness and density of subobjects in any category and classify morphisms by how they handle them, but also functors and natural transformations by how they transform them, somewhat like the theory of limits but with a simple and intuitive conceptual meaning that applies to each incarnation. We provide examples of these notions from many areas of mathematics and outline their unfolding throughout category theory as we add assumptions that bring them closer to classical settings, which we complete by generalizing the topological duality of open and closed subsets and outlining a possible framework for a categorical measure theory. Central to this interaction is the subobject 2-functor and the 2-category of upward-bounded preorders, so in the last part of the text we provide a study of their function. Concretely we derive representation theorems for this 2-functor and 2-category and a conjecture about how to generalize these theorems to n-categories.

For a normed space $E$, we denote by $E_w$ and $E'_{w^*}$ the space $E$ and its dual $E'$ endowed with the weak topology $w$ and the weak$^*$ topology $w^*$, respectively. It is well known that for a Banach space $E$, both spaces $E_w$ and $E'_{w^*}$ are Fréchet–Urysohn topological spaces if and only if $E$ is finite-dimensional. This statement remains true for all linear subspaces of $E_w$ and $E'_{w^*}$. Moreover, for any normed space $E$, every Fréchet–Urysohn subgroup of $E_w$ is locally finite-dimensional; that is, it has an open subgroup contained in a finite-dimensional linear subspace. Similarly, for any Banach space $E$, every Fréchet–Urysohn subgroup of $E'_{w^*}$ is locally finite-dimensional. Analogous results are obtained for the free locally convex space $L(X)$, its weak version $L_p(X)$, and the free topological vector space $V(X)$ over a Tychonoff space~$X$. The same conclusions hold for Baire subgroups and Baire linear subspaces of $E_w$, $E'_{w^*}$, $L(X)$, $L_p(X)$, and $V(X)$. We emphasize that even for metrizable subgroups, these results are new and nontrivial. In fact, all results are established for the much wider family of $\kappa$-Fréchet–Urysohn subgroups of topological vector spaces. The class of $\kappa$-Fréchet–Urysohn spaces is the natural framework in which the proofs of our results work.

We disprove the classical version of the Borsuk-Ulam-flavored conjecture, due to Baum-Dąbrowski-Hajac, that free actions $X\times \mathbb{G}\to X$ on non-empty compact Hausdorff spaces by non-trivial compact groups admit no equivariant maps $X*\mathbb{G}\to X$. Specifically, among non-trivial compact abelian $\mathbb{G}$, the conjecture fails precisely when $\mathbb{G}$ is torsion-free profinite.

We study bijective nonexpansive maps in $T_0$-quasi-metric spaces, distinguishing preservation of directed distances from pointwise rigidity. Our focus is their behaviour under passage to the $q$-hyperconvex hull. The upper quasi-metric integers are $q$-plastic and bicomplete, but their hull is the non-$q$-plastic upper real line. An explicit hull calculation and an established metric dense-rigidity theorem give the same failure of inheritance for a $q$-rigid original space. Tightness propagates distance preservation from the canonical copy throughout the hull. Under compactness and a bounded lattice structure for the zero-distance order, identifying that copy with the complemented elements gives a sufficient condition for inheritance of $q$-rigidity. A four-vertex asymmetric rectangle verifies this criterion. We also record metric transfer mechanisms and show that a directed interval condition gives an endpoint-coordinate representation, automatic $q$-plasticity, and $q$-rigidity under uniqueness of the ordered diametral pair.

In this paper, we investigate Lindelöf-type properties of spaces of Baire-one functions equipped with the topology of pointwise convergence. In particular, we investigate the properties of normality, extent and realcompactness for the space $B_1(X)$ of Baire-one real-valued functions. We also study when $B_1(X)$ is realcompact: we show that this is the case for every hereditarily Baire (in particular, every Polish) separable metrizable space $X$ and for every Luzin set $X$. The paper provides answers to two Teng's questions. The paper raises open questions.

We introduce a unified framework of $Q\text{--}P$ spaces, designed to encompass a broad class of generalized topological structures arising from the interaction of two topological properties. We establish general permanence and transfer results, including heredity, productivity, additivity, and separation properties. Specializing to the Hausdorff setting, we investigate the intermediate classes \(C\)-Hausdorff, \(PS\)-Hausdorff, and \(CC\)-Hausdorff, and obtain the hierarchy \[ T_2 \Longrightarrow Ps\text{-}Hausdorff \Longrightarrow CC\text{-}Hausdorff \Longrightarrow C\text{-}Hausdorff \Longrightarrow T_1. \] For \(C\)-Hausdorff spaces, we derive intrinsic characterizations and establish the equivalence with the simultaneous properties of \(k\)-Hausdorffness and \(KC\). We further examine their connections with \(k\)-spaces, Fréchet spaces, local compactness, realcompactness, and \(\aleph_0\)-boundedness. Finally, carefully constructed examples and counterexamples demonstrate the strictness of the hierarchy and the necessity of the hypothesis. Thus, the \(Q\!-\!P\) framework provides a unified perspective on generalized separation axioms and reveals a rich structure between classical Hausdorffness and \(T_1\)-separation.

We investigate how to systematically construct continuous dualities from zero-dimensional dualities, employing well-known methods from algebra, topology, category theory, and domain theory. While our method is general, this paper focusses on the move from Stone spaces to compact Hausdorff spaces and the move from Priestley spaces to compact ordered Hausdorff spaces. The engine of our approach is Stone duality for relations: on the space side quotienting by a preorder turns zero-dimensional spaces into continuous ones, while distributive lattices with a proximity relation are their algebraic duals. Our duality for relations is inherently order-enriched. Double categories organise both functional and relational morphism in the same structure. The move from zero-dimensional to continuous dualities is then a three-step construction: extend a duality from functional to relational morphism, split idempotents, restrict to maps.

A topological space is $\varkappa$-Fréchet–Urysohn if every point in the closure of an open set $U$ is the limit of a sequence of points of $U$. We prove that every $\varkappa$-Fréchet–Urysohn subgroup and every Baire subgroup of the free topological group on a Tychonoff space $X$ in any full variety of topological groups (in particular, of the free topological group $F(X)$, the free Abelian topological group $A(X)$, and the free Boolean topological group $B(X)$) is discrete. In particular, each of these groups is $\varkappa$-Fréchet–Urysohn (or Baire) only if $X$ is discrete, and all their metrizable and even Fréchet–Urysohn subgroups are discrete. Along the way we characterize strongly $\varkappa$-Fréchet–Urysohn spaces in the spirit of Michael's theorem and study countable families of closed sets absorbing convergent sequences in $\varkappa$-Fréchet–Urysohn spaces and groups.

We prove that, in most moduli spaces of orientable hyperbolic surfaces of finite area, there exist surfaces all of whose covers of degree two have a new small eigenvalue. On the other hand, we show that four punctured spheres always admit a double cover without new small eigenvalues and that there exist closed surfaces of any genus with the same property. We prove the last two of these results using a version of Cheeger's inequality adapted to double covers.

In this paper, we study the behavior of biquandle-based invariants of virtual knotoids under their connected sum. We first show that the fundamental biquandle of the connected sum of two virtual knotoids is the pushout of a span in the category of biquandles. By applying the Hom functor to this pushout description, we obtain the correspondence between biquandle colorings of $K_1\# K_2$ and compatible pairs of colorings of summands. This provides a categorical explanation of a known matrix product formula for biquandle counting matrices under connected sum. We then study the behavior of biquandle virtual bracket invariants under connected sum. We show that, for each coloring of the connected sum $K_1\#K_2$ corresponding to a compatible pair of colorings of the summands $K_1$ and $K_2$, the normalized biquandle virtual bracket value factors as the product of the normalized values of the summands. Building on this, we obtain connected-sum formulas for the normalized multiset invariants defined by utilizing biquandle virtual brackets. When the coefficient ring is a number ring, the normalized bracket multisets can be encoded by polynomials and matrices with polynomial entries. We introduce a product $\star$ on monomials and an induced matrix product $\odot$. We then show that the normalized biquandle virtual bracket matrices satisfy \[ \widetilde{\mathcal{M}}_X^{\beta}(K_1\#K_2) = \widetilde{\mathcal{M}}_X^{\beta}(K_1) \odot \widetilde{\mathcal{M}}_X^{\beta}(K_2). \]

Descriptive set theorists often refer to elements of the Baire space, $\mathcal{N} = {}^\omega\omega$, as “reals.” In this expository article, we present mathematical justification for this practice and highlight the advantages of working with $\mathcal{N}$ over $\mathbb{R}$ in the context of descriptive set theory.

We show that the numbers of colourings of a link by all finite quandles is equivalent, as an invariant of links, to the profinite completion of the peripheral system. This allows us to reformulate the classical question of the completeness of the former invariant in terms of profinite rigidity. We also develop some profinite quandle theory, and we show that a topological quandle is profinite if and only if it is compact and totally discontinuous and its group of inner automorphism is relatively compact in its homeomorphism group.

Xu asked whether the Hoare power space of a co-sober space, and in particular of a $T_2$-space or a metric space, must be co-sober. We give a negative answer. He and Zhao recently constructed a countable $T_{1}$ space $X$ whose lattice of open sets has a non-sober Scott topology. We observe that their topology has a countable subbase. The classical Ponomarev construction then yields an open continuous surjection from a countable zero-dimensional metrizable space $M$ onto $X$. Direct and inverse images then make the Scott space $\Sigma\mathcal{O}(X)$ a retract of $\Sigma\mathcal{O}(M)$, so $\Sigma\mathcal{O}(M)$ is not sober. Finally, we show that, for every $T_{0}$-space $Y$, the Hoare power space $\PH(Y)$ is co-sober if and only if the Scott space $\Sigma\mathcal{O}(Y)$ is sober. Consequently, $\PH(M)$ is not co-sober. Therefore, the same example answers negatively a question of He and Zhao about Scott sobriety of the lattices of open sets of countable Hausdorff spaces.

We introduce temporal metric spaces, a framework for formalizing metric and topological structures on continuously evolving domains. Rather than defining a single metric on a fixed space, we construct a family of interval-dependent metrics on an associated trajectory space, enabling distances to be measured between continuous trajectories over finite time intervals. We establish the fundamental topology of the trajectory space, prove that it inherits completeness from the underlying metric space, and develop the corresponding temporal hyperspace together with its completeness. Finally, we develop a theory of iterated function systems acting on these evolving spaces, proving that their attractors evolve coherently with the underlying dynamics and remain stable even when the evolving geometry causes the generating mappings to lose their classical contraction property.