Quasi-Open Map¶

Map every nonempty open subset of a topological domain to a set with nonempty interior, retaining a weakened openness guarantee without requiring the image itself to be open.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2601
Origin domain: mathematics
Subdomain: general topology
Aliases: Quasi-interior map, Quasi-open mapping

Core Idea¶

A quasi-open map is a function \(f:X\to Y\) between topological spaces such that, for every nonempty open subset \(U\subseteq X\), the image \(f(U)\) has nonempty interior in \(Y\). Equivalently, \(\operatorname{int}_Y f(U)\neq\varnothing\) whenever \(U\) is open and nonempty. The condition weakens openness: an open map requires every \(f(U)\) to be open, whereas a quasi-open map permits a nonopen image so long as it contains some nonempty open part. Mardešić and Papić use this property in their study of continuous images of ordered compacta and dyadic compacta, where it preserves enough local largeness to support global topological arguments.^[1]

The word map does not silently add continuity. Under the definition used here, quasi-openness and continuity are independent predicates that may be imposed together in a theorem. Every open map is quasi-open, but the converse fails. For example, \(f:\mathbb R\to\mathbb R\) given by \(f(x)=x^2\) is continuous and quasi-open but not open: every nonempty open set contains an interval on which the image has nonempty interior, yet \(f((-1,1))=[0,1)\) is not open in \(\mathbb R\). Conversely, a discontinuous function can satisfy the image-interior condition. Kao's treatment of quasi-open maps in the theory of \(M_1\)-spaces demonstrates that the property is a recognized mapping condition rather than informal language for an almost-open function.^[2]

Quasi-openness is useful because it rules out the total collapse of a nonempty open region into an image with empty interior. It is closed under composition: if each of two maps sends every nonempty open set to a set containing a nonempty open set, then their composite does also. Surjectivity is not part of the definition, although particular results may assume it. Nor does the condition control fibers, separation axioms, compactness, quotient behavior, or preservation of arbitrary unions beyond what follows from ordinary image formation. The stable abstraction is precisely the open-set image predicate and its consequences, not every theorem in which quasi-open maps occur.

Structural Signature¶

Topological domain and codomain. The source and target carry explicit topologies.
Function. A single-valued assignment \(f:X\to Y\) supplies the image operation.
Universal open-set quantifier. Every nonempty open subset of the domain is tested.
Image formation. The whole set-theoretic image of each tested subset is considered.
Target-relative interior. Interior is computed in the topology of the codomain, including when the codomain is a subspace.
Nonvanishing condition. The required interior is nonempty, not necessarily equal to the image.
No built-in continuity. Continuity is a separate hypothesis unless explicitly added.
No built-in surjectivity. The property concerns images of open sets, not coverage of all target points.
Open-map weakening. Full openness implies the condition, while the residual allows nonopen images.
Compositional stability. Composites of quasi-open maps remain quasi-open.
Local-largeness preservation. Nonempty open information cannot be mapped entirely into a set with empty interior.
Convention declaration. Results must state whether continuity or surjectivity is included by local authorial convention.

What It Is Not¶

Not an open map. The image need only contain a nonempty open part; it need not itself be open.
Not automatically continuous. The image condition does not imply inverse-image preservation of open sets.
Not automatically surjective. A proper target region may contain all images.
Not a quotient map. Quotient topology is defined by inverse images and generally requires continuity and surjectivity.
Not a closed map. No condition is imposed on images of closed subsets.
Not an almost-open map under every convention. Neighboring terminology varies and must be defined rather than assumed synonymous.
Not a density claim. Nonempty interior is stronger than nonemptiness and different from density.
Not a property of one favored open set. The quantifier ranges over every nonempty open subset of the source.

Scope of Application¶

Quasi-openness is literal in general topology whenever a theorem needs images of locally substantial source regions to retain some target interior, while full image openness would be unnecessarily strong.

Continuous-image theorems. Continuity and quasi-openness can jointly transfer selected covering or compactness-related properties.
Ordered compacta. The condition appears in analyses of continuous images of ordered compact spaces.
Generalized metric spaces. Mapping results use quasi-openness to study preservation of network and base properties.
Continuum theory. Surjective continuous quasi-open maps provide a controlled middle ground between arbitrary continuous maps and open maps.
Topological dynamics. Factor maps may be checked for quasi-openness when open-set largeness matters.
Composition arguments. A chain of mappings retains the property without requiring each intermediate image to be open.
Counterexample design. The distinction separates consequences of continuity, openness, surjectivity, and quotient structure.
Subspace codomains. Relative interior prevents errors caused by computing interior in an ambient space instead of the declared target.

Clarity¶

Declare the topologies on \(X\) and \(Y\), the function \(f\), and whether continuity or surjectivity is separately assumed. State the quantifiers exactly: for every nonempty open \(U\subseteq X\), the target-relative interior \(\operatorname{int}_Y f(U)\) must be nonempty. Do not replace this with the weaker assertion that images are nonempty, the different assertion that images are dense, or the stronger assertion that images are open. If a codomain is a subspace, say so before computing interior. When citing a result whose author defines quasi-open map to include continuity or surjectivity, translate that convention explicitly. To disprove quasi-openness, exhibit one nonempty open source set whose image has empty target interior. To prove it, avoid verifying only basis elements unless the basis reduction argument is stated.

Manages Complexity¶

The abstraction compresses a family of open-set image tests into one reusable mapping predicate. It lets a proof carry forward the fact that no locally visible part of the source becomes topologically negligible in the target, without paying for full openness. This makes theorem hypotheses more precise and reveals which conclusions rely on continuity, surjectivity, compactness, or separation axioms instead. The compression can mislead if local conventions bundle those extra hypotheses into the name. It also says little about how large the retained interior is, where it lies, whether fibers are small, or how boundaries behave. Quasi-openness manages one dimension of complexity—preservation of nonempty interior—and leaves the rest explicit.

Abstract Reasoning¶

Specify source and target topologies and the exact function.
Separate continuity, surjectivity, compactness, and separation assumptions from quasi-openness.
Choose an arbitrary nonempty open subset of the source.
Compute or characterize its full image in the codomain.
Take interior relative to the declared codomain topology.
Prove that this interior is nonempty without assuming the desired result.
Discharge the universal quantifier, possibly through a justified basis reduction.
For compositions, select an open subset inside the first image and apply the second map's property.
For counterexamples, isolate one open source set whose image is thin in the target.
Translate the result back into the theorem's local terminology and conventions.

Knowledge Transfer¶

The strict parent is Function Mapping. Quasi-openness classifies functions by a rule relating open subsets of their domain to the interiors of their images in the codomain. Function Mapping supplies the substrate-independent input-to-output assignment and image operation; topology supplies the domain accent. The property transfers to any topological substrate but does not become a prime because its invariant depends essentially on open sets and interior. Open Map, Continuous Map, and Quotient Map remain neighboring domain-specific combinations of different predicates.

Examples¶

Canonical¶

Let \(f:\mathbb R\to\mathbb R\) be \(f(x)=x^2\). Every nonempty open set contains a smaller interval on which \(f\) is nonconstant and monotone, so its image contains a nonempty open interval. Thus \(f\) is quasi-open. It is not open because \(f((-1,1))=[0,1)\), which is not open in \(\mathbb R\). The example separates the nonempty-interior guarantee from full openness without relying on a change of codomain topology.

Mapped back: topological function + arbitrary nonempty open source region → image → nonempty target-relative interior, without requiring the whole image to be open.

Applied / In Practice¶

A theorem factors a continuous surjection through two intermediate spaces. The first factor is known only to be quasi-open and sends a selected open neighborhood to a set containing an open target patch. The second factor is also quasi-open, so applying it to that patch yields an image with nonempty interior in the final space. The composite therefore retains quasi-openness even though neither intermediate image is known to be open. The proof records continuity and surjectivity separately rather than hiding them in the label.^[1]

Mapped back: factorized mapping → open patch retained after first image → open patch retained after second image → composite nonempty-interior guarantee.

Structural Tensions¶

Weakened openness vs. useful control. Nonempty interior preserves largeness but not image openness. Diagnostic: Does the theorem need an open image or merely an open subset inside it?
Definition vs. convention. Some authors bundle continuity or surjectivity into the term. Diagnostic: Which predicates are independently stated in the cited source?
Relative vs. ambient interior. A subspace image can be internally open while ambiently thin. Diagnostic: In which topology is interior computed?
Basis checking vs. universal checking. A basis can simplify proof only with a valid containment argument. Diagnostic: Does every nonempty open set contain a tested basis element whose image has interior?
Composition vs. arbitrary image nesting. The compositional proof depends on finding an open subset inside the first image. Diagnostic: Is the second quasi-open predicate applied to an actually open source subset?
Autonomous property vs. Function Mapping. Every quasi-open map is a function, but the open-image predicate is additional. Diagnostic: Does the candidate retain a theorem-bearing residual beyond generic assignment?

Structural–Framed Character¶

The function, source and target topologies, universal nonempty-open-set quantifier, image, and target-relative nonempty interior are structural. Continuity, surjectivity, compactness, connectedness, separation axioms, and the theorem in which the property is used are framed additional assumptions. The property certifies only that each nonempty open source region has an image with some interior. It does not quantify that interior, ensure the whole image is open, or warrant geometric language such as locally volume preserving.

Structural Core vs. Domain Accent¶

The transferable skeleton is a function constrained by a universal subset-to-image predicate. The domain accent is topological openness and interior, including relative topology and the distinction from inverse-image continuity. Removing that accent yields Function Mapping; strengthening the residual to image openness yields Open Map rather than Quasi-Open Map.

Function Mapping is the strict parent because a quasi-open map is a function whose images satisfy an additional topological predicate. The proposed composition/presupposes relation records the required mapping substrate without claiming that every function is topological or quasi-open.

The prospective workspace queue contains one strict upward edge to prime:function_mapping. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Quasi-Open Map Domain-specific

Parents (1) — more general patterns this builds on

Quasi-Open Map is a kind of Function (Mapping) Prime

Function Mapping is the strict parent because a quasi-open map is a function whose images satisfy an additional topological predicate.

Hierarchy path (1) — routes to 1 parentless root

Quasi-Open Map → Function (Mapping)

Neighborhood in Abstraction Space¶

Quasi-Open Map sits in a sparse region of the domain-specific corpus (76^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Functions, Maps & Integral Structure (10 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Open Map. Sends every open set to an open image, a strictly stronger requirement.
Continuous Map. Constrains inverse images of open sets, not interiors of direct images.
Quotient Map. A surjective continuous map determining target openness through inverse images.
Closed Map. Sends closed sets to closed images.
Almost Open Map. A variable term whose definition must be checked in its source.
Interior Map in modal logic. A different construction combining topological continuity and openness.
Dense Image. A global property of the range, not a universal local image-interior condition.

References¶

[1] Sibe Mardešić and Pavle Papić, “Continuous Images of Ordered Compacta, the Suslin Property and Dyadic Compacta,” Glasnik Matematičko-Fizički i Astronomski, Series II 17 (1962): 3–22, https://web.math.pmf.unizg.hr/glasnik/skenirano/mardesicpapic1962.pdf. registry ↩a ↩b

[2] Kuo Shih Kao, “A Note on Quasi-Open Maps,” Pacific Journal of Mathematics 108, no. 1 (1983): 121–128, https://doi.org/10.2140/pjm.1983.108.121. registry ↩