Soft Usual Topology

• Soft Usual Topology is a framework that extends the classical topology of ℝ into soft sets by parameterizing open sets via an auxiliary index set.
• It distinguishes between parameter-wise and single-set constructions, highlighting differences in the transfer and preservation of separation, regularity, and connectedness.
• The topology aligns each parameter with the usual topology of ℝ, enabling the study of soft analogues for properties such as connectedness, path-connectedness, and continuity.

The soft usual topology is a canonical soft topological structure that extends the classical topology of the real numbers $\mathbb{R}$ into the soft set framework, parametrizing open sets via an auxiliary index set. This construction yields a soft topological space that is aligned with the usual topology on $\mathbb{R}$ in every parameter, enabling the study of soft analogues of classical topological properties such as connectedness, path-connectedness, and separation axioms. Two principal constructions—parameter-wise and constant (single-set) soft topologies—organize the landscape, providing a nuanced comparison between soft and crisp topology and highlighting subtleties in the transfer and preservation of topological properties.

1. Soft Set and Soft Topological Space Structures

Let $X$ be a nonempty set and $E$ a nonempty set of parameters. A soft set over $X$ is a mapping $F: E \to \mathcal{P}(X)$, written as $(F,E) = \{(e, F(e)) : e \in E\}$. The collection of all such is $S_E(X)$. A soft topology $\Sigma \subseteq S_E(X)$ must contain the null soft set (where each fiber is $\varnothing$) and the absolute soft set (each fiber is $X$), be stable under finite soft intersections, and under arbitrary soft unions.

Given a soft topological space $(X, E, \Sigma)$, each $e \in E$ specifies a “crisp” topology $\Sigma_e = \{ F(e) : (F,E) \in \Sigma \}$ on $X$ (Ameen et al., 2023).

2. Definition and Construction of the Soft Usual Topology

For the real numbers $\mathbb{R}$ with their usual topology, and a nonempty parameter set $\xi$, the soft usual topology $\mathscr{U}_\xi$ is constructed by:

• Soft Subbasis: Families $M^a_\xi$, $N^b_\xi$ for $a, b \in \mathbb{R}$ are defined by $M^a_\xi(e) = (a,\infty)$ and $N^b_\xi(e) = (-\infty, b)$, for each $e \in \xi$. The soft subbasis is $$\mathscr{S}_\xi = \{ M^a_\xi \mid a \in \mathbb{R} \} \cup \{ N^b_\xi \mid b \in \mathbb{R} \}$$.
• Soft Basis: The family $$\mathscr{B}_\xi = \{ H^{(a,b)}_\xi \mid a < b; H^{(a,b)}_\xi(e) = (a,b) \}$$.
• Full Soft Topology: Arbitrary soft unions of $\mathscr{B}_\xi$ yield all sets of the form $H^G_\xi(e) = G$, $G \subset \mathbb{R}$ open in the usual sense. Thus, $$\mathscr{U}_\xi = \{ H^G_\xi \mid G \text{ open in the usual topology} \}$$ (Alemdar et al., 16 Nov 2025).

The soft usual topological space is $(\mathbb{R}, \mathscr{U}_\xi)_\xi$.

3. Parameter-wise and Single-set Soft Usual Topologies

Two explicit constructions for soft topologies on $(X, E)$, both yielding versions of the “soft usual topology,” are distinguished (Ameen et al., 2023):

Construction	Definition in LaTeX	Description
Parameter-wise ($\Sigma_1$)	$\{(F, E):\;F(e)\in\tau\;\forall e\}$	Each fiber $F(e)$ is a usual open set
Single-set ($\Sigma_2$)	$\{(U, E):\;U\in\tau\}$	All fibers are equal to the same open set

Here, $\tau$ is the usual (crisp) topology on $X$. For $|E| \geq 2$ and $\tau$ nontrivial, $\Sigma_2 \subsetneqq \Sigma_1$. $\Sigma_2$ is isomorphic to the original topology $\tau$ via $U \mapsto (U, E)$; $\Sigma_1$ is strictly finer when $|E| > 1$ (Ameen et al., 2023).

4. Compatibility and Topological Properties

For each parameter $e \in \xi$, the “$e$-section” $$(\mathscr{U}_\xi)^e = \{ W(e) : W \in \mathscr{U}_\xi \}$$ recovers the usual topology on $\mathbb{R}$ (Alemdar et al., 16 Nov 2025). Thus, all topological properties present in $(\mathbb{R}, \tau)$—Hausdorffness, connectedness, path-connectedness—are present “softly,” i.e., for every parameter.

Key properties:

• Soft Hausdorff: For $x \neq y \in \mathbb{R}$, there exist disjoint soft opens separating $x$ and $y$.
• Soft Connected: Any soft separation induces a crisp separation, impossible for connected spaces like $\mathbb{R}$.
• Soft Path Connected: Standard paths $\gamma(t) = (1-t)a + tb$ induce soft paths $\Gamma$.

The soft subspace topology on $[0,1]$ is inherited in a parameter-wise manner, ensuring that the soft versions of connectedness and path-connectedness are preserved by this restriction (Alemdar et al., 16 Nov 2025).

5. Soft Open Sets: Examples and Characterization

Soft open sets in the soft usual topology are “absolute”: for each parameter $e \in \xi$, the value is a usual open set in $\mathbb{R}$. For example, with $\xi = \{e_1, e_2, e_3\}$ and $G = (-3,2)$, $$H^{(-3,2)}_\xi = \{(e_1, (-3,2)), (e_2, (-3,2)), (e_3, (-3,2))\}$$. The whole space is $$H^\mathbb{R}_\xi = \{(e_1, \mathbb{R}), (e_2, \mathbb{R}), (e_3, \mathbb{R})\}$$ (Alemdar et al., 16 Nov 2025).

In $\Sigma_1$, a much richer soft open structure is allowed, with potentially differing open sets across parameters, unless constant fibers are enforced (yielding $\Sigma_2$).

6. Separation and Regularity: Soft Topological Axioms

Both $\Sigma_1$ and $\Sigma_2$ inherit soft $T_0$, $T_1$, and $T_2$ (soft Hausdorffness) from the usual topology. In $\Sigma_2$, all higher separation properties such as soft regularity and soft normality are preserved if the base topology is regular or normal, as $\Sigma_2$ is simply a relabeling of the original crisp topology.

In contrast, $\Sigma_1$ need not be soft regular or soft normal when $|E| > 1$. For example, in $\mathbb{R}$, with $E = \{e_1, e_2\}$, define a soft closed set $(F, E)$ by $F(e_1) = [0,1]$, $F(e_2) = \mathbb{R}$. The standard proof by contradiction shows that $(\mathbb{R}, E, \Sigma_1)$ is not soft regular, despite each fiber being so (Ameen et al., 2023).

7. Soft Continuity and Morphisms

A soft mapping $$(\mathbb{I}_\xi, \varrho): (X, \zeta)_\xi \to (X', \zeta')_\xi$$ is soft continuous at $x$ if, for each soft open $W' \in \zeta'$ containing $\varrho(x)$, there exists a soft open $W \in \zeta$ about $x$ with $\varrho(W) \,\widetilde{\sqsubseteq}\, W'$. Equivalently, continuity is characterized by the preimage of every soft open set being soft open (Alemdar et al., 16 Nov 2025).

For instance, the map $\gamma(t) = 2t$ from $I = [0,1]$ to $\mathbb{R}$ induces a soft mapping $$\Gamma = (\mathbb{I}_\xi, \gamma): (I, (\mathscr{U}_\xi)_I) \to (\mathbb{R}, \mathscr{U}_\xi)$$, soft continuous by fiberwise $\varepsilon$–$\delta$ arguments.

These mapping notions extend to the study of soft topological groups and the corresponding category-theoretic structures, including symmetric monoidal categories (Alemdar et al., 16 Nov 2025).

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The soft usual topology, constructed by Alemdar, Akız, and Ayaz, and analyzed in detail by Ameen, Asaad, and Mohammed, provides a natural soft analogue to classical topology on $\mathbb{R}$. It preserves the traditional topological properties in a soft context while offering new degrees of freedom for parameterization and generalization. The distinction between parameter-wise and single-set constructions traces to foundational differences in the preservation of regularity, normality, and other higher separation axioms, making the study of the soft usual topology central to understanding the broader behavior of soft topological spaces (Alemdar et al., 16 Nov 2025, Ameen et al., 2023).