A mathematical discipline assembling the topology and group is called topological group. This has very significant applications in almost all branches of natural sciences. In our arrangement operations multiplicity inverse on continuity its general forms will be discussed. The study this weaker form with groups started 1990s. Twenty-thirty years ago more interesting results relating to discusse...

1 Topological Spaces 1 1.1 Continuity and Topological Spaces . . . . . . . . . . . . . . . 1 1.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Further Examples of Topological Spaces . . . . . . . . . . . . 3 1.5 Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Hausdorff ...

In this paper by definition of generalized action of generalized Lie groups (top spaces) on a manifold, the concept of stabilizer of the top spaces is introduced. We show that the stabilizer is a top space, moreover we find the tangent space of a stabilizer. By using of the quotient spaces, the dimension of some top spaces are fined.

1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology 20 2.4. Gluing topologies 23 2.5. Proper maps 25 3. Connectedness 26 4. Separation axioms and the Hausdorff property 32 4.1. More on the Hausdorff property 34 5. Compactness and its relatives 35 5.1. Local compactness and paracompactness 41 5.2. Compactness in metric spaces...

The properties of the metric topology on infinite and finite sets are analyzed. We answer whether finite metric spaces hold interest in algebraic topology, and how this result is generalized to pseudometric spaces through the Kolmogorov quotient. Embedding into Lebesgue spaces is analyzed, with special attention for Hilbert spaces, `p, and EN .

The main purpose of this paper is to establish general conditions under which T2-spaces are compact-covering images of metric spaces by using the concept of cfpcovers. We generalize a series of results on compact-covering open images and sequencecovering quotient images of metric spaces, and correct some mapping characterizations of g-metrizable spaces by compact-covering σ-maps and mssc-maps.

The quotient space theory based on fuzzy tolerance relation is put forward to solve the problem of clustering in this paper. The similarity matrix does not always satisfy ultrametric inequality, theoretically and practically. We give the method to construct the hierarchical quotient space chain if the similarity matrix is only reflexive and symmetric. We consider not only the subset of data (cl...

Example 1.1 (Cosets in R). Consider the vector space X = R. Let M be any onedimensional subspace of R, i.e., M is a line in R through the origin. A coset of M is a rigid translate of M by a vector in R. For concreteness, let us consider the case where M is the x1-axis in R , i.e., M = {(x1, 0) : x1 ∈ R}. Then given a vector y = (y1, y2) ∈ R , the coset y +M is the set y +M = {y +m : m ∈ M} = {(...