Open, Connected Functions

1973 ◽  
Vol 16 (1) ◽  
pp. 57-60 ◽  
Author(s):  
Louis Friedler
Keyword(s):  
Monotone Function ◽  
Connected Subset ◽  
Connected Space ◽  
Multivalued Functions ◽  
Connected Spaces

Recall that a function f:X→ Yis called connected if f(C) is connected for each connected subset C of X. These functions have been extensively studied. (See Sanderson [6].) A function f:X → Y is monotone if for each y ∊ Y, f-1(y) is connected. We shall use the techniques of multivalued functions to prove that if f: X→ Y is open and monotone onto Y, then f-1(C) is connected for each connected subset C of Y. This result is used to prove that the product of semilocally connected spaces is semilocally connected and that the image of a maximally connected space under an open, connected, monotone function is maximally connected.

scholarly journals New types of locally connected spaces via clopen set

2021 ◽  
Vol 40 (3) ◽  
pp. 671-679
Author(s):  
Ennis Rafael Rosas Rodríguez ◽  
Sarhad F. Namiq
Keyword(s):  
Connected Space ◽  
New Type ◽  
Connected Spaces

In this paper, we define and study a new type of connected spaces called λco-connected space. It is remarkable that the class of λ-connected spaces is a subclass of the class of λco-connected spaces. We discuss some characterizations and properties of λco-connected spaces, λco components and λco-locally connected spaces

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions II

1992 ◽  
Vol 122 (1-2) ◽  
pp. 127-135 ◽  
Author(s):  
John W. Rutter
Keyword(s):  
Group Extension ◽  
Equivalence Classes ◽  
Homotopy Groups ◽  
Geometric Proof ◽  
Homotopy Equivalence ◽  
Connected Space ◽  
Abelian Kernel ◽  
Simply Connected ◽  
Extension Sequence ◽  
Connected Spaces

SynopsisWe give here an abelian kernel (central) group extension sequence for calculating, for a non-simply-connected space X, the group of pointed self-homotopy-equivalence classes . This group extension sequence gives in terms of , where Xn is the nth stage of a Postnikov decomposition, and, in particular, determines up to extension for non-simplyconnected spaces X having at most two non-trivial homotopy groups in dimensions 1 and n. We give a simple geometric proof that the sequence splits in the case where is the generalised Eilenberg–McLane space corresponding to the action ϕ: π1 → aut πn, and give some information about the class of the extension in the general case.

scholarly journals Pythagorean Fuzzy Rough Connected Spaces

2021 ◽  
Vol 2070 (1) ◽  
pp. 012069
Author(s):  
P Revathi ◽  
R Radhamani
Keyword(s):  
Rough Set ◽  
Topological Spaces ◽  
Connected Space ◽  
Fuzzy Rough Set ◽  
Connected Spaces

Abstract In this paper Pythagorean fuzzy rough set and Pythagorean fuzzy rough topological spaces are defined for the connected space. Then, the properties of connectedness are discussed with examples.

scholarly journals Almost locally connected spaces

1981 ◽  
Vol 31 (4) ◽  
pp. 421-428 ◽  
Author(s):  
Vincent J. Mancuso
Keyword(s):  
Connected Space ◽  
Local Connectedness ◽  
Quotient Maps ◽  
Regular Open ◽  
Connected Spaces ◽  
Open Maps

AbstractThis paper introduces the concept of an almost locally connected space. Every locally connected space is almost locally connected, and the concepts are equivalent in the class of semi-regular spaces. Almost local connectedness is hereditary for regular open subspaces, is preserved by continuous open maps, but not generally by quotient maps. It is productive in the presence of almost-regularity.

Order-continuous functions and order-connected spaces

1970 ◽  
Vol 68 (1) ◽  
pp. 27-31 ◽  
Author(s):  
D. C. J. Burgess ◽  
S. D. McCartan
Keyword(s):  
Continuous Functions ◽  
Relative Importance ◽  
Connected Space ◽  
Order Continuity ◽  
Ordered Space ◽  
Order Continuous ◽  
Connected Spaces

We introduce and compare four procedures for defining the order-continuity of a function from one topological ordered space into another, where each reduces to the usual conception when the orderings of the two spaces are trivial. Chiefly for the purposes of this comparison, we use the idea of an ‘order-connected’ space, and in the course of investigating under which types of order-continuous functions this property is preserved, we are helped in assessing their relative importance.

scholarly journals Co-H-Spaces and Almost Localization

2014 ◽  
Vol 58 (2) ◽  
pp. 323-332
Author(s):  
Cristina Costoya ◽  
Norio Iwase
Keyword(s):  
Universal Covering ◽  
Connected Space ◽  
Converse Statement ◽  
Simply Connected ◽  
Connected Spaces

AbstractApart from simply connected spaces, a non-simply connected co-H-space is a typical example of a space X with a coaction of Bπ1 (X) along rX: X → Bπ1 (X), the classifying map of the universal covering. If such a space X is actually a co-H-space, then the fibrewise p-localization of rX (or the ‘almost’ p-localization of X) is a fibrewise co-H-space (or an ‘almost’ co-H-space, respectively) for every prime p. In this paper, we show that the converse statement is true, i.e. for a non-simply connected space X with a coaction of Bπ1 (X) along rX, X is a co-H-space if, for every prime p, the almost p-localization of X is an almost co-H-space.

scholarly journals Returning functions with closed graph are continuous

2020 ◽  
Vol 70 (2) ◽  
pp. 297-304
Author(s):  
Taras Banakh ◽  
Małgorzata Filipczak ◽  
Julia Wódka
Keyword(s):  
Topological Space ◽  
Real Number ◽  
Positive Real Number ◽  
Connected Subset ◽  
Closed Graph ◽  
Positive Real ◽  
Contractible Spaces ◽  
Path Connected ◽  
Connected Spaces

Abstract A function f : X → ℝ defined on a topological space X is called returning if for any point x ∈ X there exists a positive real number Mx such that for every path-connected subset Cx ⊂ X containing the point x and any y ∈ Cx ∖ {x} there exists a point z ∈ Cx ∖ {x, y} such that |f(z)| ≤ max{Mx, |f(y)|}. A topological space X is called path-inductive if a subset U ⊂ X is open if and only if for any path γ : [0, 1] → X the preimage γ–1(U) is open in [0, 1]. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible spaces. We prove that a function f : X → ℝ defined on a path-inductive space X is continuous if and only if it is returning and has closed graph. This implies that a (weakly) Świątkowski function f : ℝ → ℝ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscribed to Lviv Scottish Book.

scholarly journals Infra -α- Compact and Infra -α- Connected Spaces

2021 ◽  
Vol 8 ◽  
pp. 41-57
Author(s):  
Raja Mohammad Latif
Keyword(s):  
Compact Space ◽  
General Topology ◽  
Topological Spaces ◽  
Connected Space ◽  
Fundamental Properties ◽  
Open Set ◽  
New Concepts ◽  
The Relationship ◽  
Connected Spaces

In 2016 Hakeem A. Othman and Md. Hanif Page introduced a new notion of set in general topology called an infra -α- open set and investigated its fundamental properties and studied the relationship between infra -α- open set and other topological sets. The objective of this paper is to introduce the new concepts called infra -α- compact space, countably infra -α- compact space, infra -α- Lindelof space, almost infra -α- compact space, mildly infra -α- compact space and infra -α- connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.

Connected Maps and Essentially Connected Spaces

1985 ◽  
Vol 28 (2) ◽  
pp. 190-194
Author(s):  
Eizo Nishiura
Keyword(s):  
Monotone Mappings ◽  
Connected Space ◽  
Continuity Properties ◽  
Weak Monotonicity ◽  
Connected Spaces

AbstractThe paper discusses some consequences of weak monotonicity for connected maps in relation to essential connectedness of a space. The first main result gives conditions under which the image by a connected map of an essentially connected space is essentially connected. The second is that, for a connected mapping of a connected, 1 .c. space to a WLOTS-wise and essentially connected space, w-monotonicity implies monotonicity. The remainder of the paper discusses continuity properties of connected, w-monotone mappings with WLOTS-wise and essentially connected range.

scholarly journals Some Results on mX-N-connected Space

2021 ◽  
pp. 604-612
Author(s):  
Ahmed A. Salih ◽  
Haider J. Ali
Keyword(s):  
Connected Space ◽  
Open Set ◽  
Connected Spaces

In this essay, we utilize m - space to specify mX-N-connected, mX-N-hyper connected and mX-N-locally connected spaces and some functions by exploiting the intelligible mX-N-open set. Some instances and outcomes have been granted to boost our tasks.

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