local and global convexity for maps
... 1. Theorem (Tietze-Nakajima). Let X be a closed, connected, and locally convex subset of Rn . Then X is convex. 2. Example. A disjoint union of two closed balls is closed and locally convex but is not connected. A punctured disk is connected and satisfies the locally convexity condition but it is no ...
... 1. Theorem (Tietze-Nakajima). Let X be a closed, connected, and locally convex subset of Rn . Then X is convex. 2. Example. A disjoint union of two closed balls is closed and locally convex but is not connected. A punctured disk is connected and satisfies the locally convexity condition but it is no ...
Topology - University of Nevada, Reno
... 1.3. Continuity and convergence in terms of open and closed sets The purpose of this section is to prove that the continuity of a function f : Rn → Rm either at a point or globally, can be expressed entirely in terms of open or closed sets. Recall that the preimage f −1 (V ) of a function f : X → Y ...
... 1.3. Continuity and convergence in terms of open and closed sets The purpose of this section is to prove that the continuity of a function f : Rn → Rm either at a point or globally, can be expressed entirely in terms of open or closed sets. Recall that the preimage f −1 (V ) of a function f : X → Y ...
Elementary Topology Problem Textbook O. Ya. Viro, OA
... A set is determined by its elements. The set is nothing but a collection of its elements. This manifests most sharply in the following principle: two sets are considered equal if and only if they have the same elements. In this sense, the word set has slightly disparaging meaning. When something is ...
... A set is determined by its elements. The set is nothing but a collection of its elements. This manifests most sharply in the following principle: two sets are considered equal if and only if they have the same elements. In this sense, the word set has slightly disparaging meaning. When something is ...
Thom Spectra that Are Symmetric Spectra
... symmetric Thom spectrum functor shows that the category I is closely related to the category of symmetric spectra. However, many of the Thom spectra that occur in the applications do not naturally arise from a map of I-spaces but rather from a map of D-spaces for some monoidal category D equipped wi ...
... symmetric Thom spectrum functor shows that the category I is closely related to the category of symmetric spectra. However, many of the Thom spectra that occur in the applications do not naturally arise from a map of I-spaces but rather from a map of D-spaces for some monoidal category D equipped wi ...
sA -sets and decomposition of sA
... 2. locally closed [2] if A = L ∩ M, where L is open and M is closed in X. 3. semi-I-regular [14] if A is a t-I-set and semi-I-open in X. Definition 1.12. [9] An ideal space (X, τ , I) is called I-submaximal if every ⋆-dense subset of X is open in X. Definition 1.13. [7] An ideal space (X, τ , I) is ...
... 2. locally closed [2] if A = L ∩ M, where L is open and M is closed in X. 3. semi-I-regular [14] if A is a t-I-set and semi-I-open in X. Definition 1.12. [9] An ideal space (X, τ , I) is called I-submaximal if every ⋆-dense subset of X is open in X. Definition 1.13. [7] An ideal space (X, τ , I) is ...
Model Categories and Simplicial Methods
... in C. While this is easy, note that the fibrant objects have changed: q : X → A in C/A is fibrant if and only if q is a fibration in C. Example 1.8. Another basic example of a model category is the category of topological spaces, Top. A continuous map f : X −→ Y is a weak equivalence if f∗ : πk (X, ...
... in C. While this is easy, note that the fibrant objects have changed: q : X → A in C/A is fibrant if and only if q is a fibration in C. Example 1.8. Another basic example of a model category is the category of topological spaces, Top. A continuous map f : X −→ Y is a weak equivalence if f∗ : πk (X, ...
On Chains in H-Closed Topological Pospaces
... In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [3, 4, 7–10, 14, 17]. If A is a subset of a topological space X, then we denote the closure of the set A in X by cl X (A). By a partial order on a set X we mean a reflexive, transitive and anti-s ...
... In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [3, 4, 7–10, 14, 17]. If A is a subset of a topological space X, then we denote the closure of the set A in X by cl X (A). By a partial order on a set X we mean a reflexive, transitive and anti-s ...
Lecture 1: August 25 Introduction. Topology grew out of certain
... Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, “the motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are ...
... Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, “the motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are ...
On functions between generalized topological spaces - RiuNet
... between generalized topological spaces. Specifically, if τX and τY are (ordinary) topologies on X and Y , then (τX , τY )-continuity is classical topological continuity. Furthermore, (sτX , τY )-continuity is the semi-continuity of [10], (pτX , τY )-continuity is precontinuity in the sense of [11], ...
... between generalized topological spaces. Specifically, if τX and τY are (ordinary) topologies on X and Y , then (τX , τY )-continuity is classical topological continuity. Furthermore, (sτX , τY )-continuity is the semi-continuity of [10], (pτX , τY )-continuity is precontinuity in the sense of [11], ...
characterizations of feebly totally open functions
... (X, τ), then the intersection of all semi-closed sets containing A is called the seni-closure of A, and is denoted by sCl(A). The largest semiopen set contained in A is denoted by sInt(A). In 1963 Levine [28] showed that since the semiopen set of a topological space need not be closed under finite i ...
... (X, τ), then the intersection of all semi-closed sets containing A is called the seni-closure of A, and is denoted by sCl(A). The largest semiopen set contained in A is denoted by sInt(A). In 1963 Levine [28] showed that since the semiopen set of a topological space need not be closed under finite i ...