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• A topological space is a set endowed with a topology.
• The elements of the underlying set of a topological space are termed points, and are typically denoted by lower case letters.
• Subsets of a topological space are typically denoted by capital letters.
• Thus, a collection of open subsets of a topological space might be denoted where is an indexing set.
• Likewise, the concept of a topological space is concerned with generalizing the structure of sets in Euclidean spaces.
• Of course, for many topological spaces the similarities are remote, but aid in judgment and guide proofs.
• In fact, there are many equivalent ways to define what we will call a topological space just by defining families of subsets of a given set.
• The properties of the topological space depend on the number of subsets and the ways in which these sets overlap.
• The book first offers information on elementary principles, topological spaces, and compactness and connectedness.
• We introduce the notion of generalized topological space (gt-space).
• Such axioms limit the size of the topology in a way, and are often satisfied by important topological spaces that occur in applications.
• Properties that are defined for a topological space can be applied to a subset of the space, with the relative topology.
• Suppose that \( S \) has the trivial topology and that \( (T, \mathscr{T}) \) is another topological space.
• Suppose that \( S \) has the discrete topology and that \( (T, \mathscr{S}) \) is another topological space.
• A topological space can be used to define a topology on any particular subset or on another set.
• A topological space in which the points are functions is called a function space.
• There are many other equivalent ways to define a topological space.
• A variety of topologies can be placed on a set to form a topological space.
• A function between topological spaces is said to be continuous if the inverse image of every open set is open.
• The basis of topological space contains cylindrical open sets.
• Sections 2–3 develop standard facts, mostly elementary, about how certain combinations of properties of topological spaces imply others.
• Sequences by themselves are inadequate for detecting convergence in general topological spaces, and nets are a substitute.
• An ideal topological space (or ideal space) means a topological space with an ideal defined on .
• Let be a topological space with an ideal defined on .
• Let be a subset of an ideal topological space .
• In this paper, we introduce the notion of expanding topological space.
• In particular, we prove that the more a topological space expands, the finer the topology of its indexed states is.
• When regarding a base of an open, or closed, topology, it is common to refer to it as an open or closed base of the given topological space.
• Open bases are more often considered than closed ones, hence if one speaks simply of a base of a topological space, an open base is meant.
• cardinal number that is the cardinality of a base of a given topological space is called its weight (cf.
• This approach was chosen by K. Kuratowski (1922) in order to construct the concept of a topological space.
• For the mathematical structure, see Topological space .
• If τ is a topology on X, then the pair (X, τ) is called a topological space.
• A function or map from one topological space to another is called continuous if the inverse image of any open set is open.
• A manifold is a topological space that resembles Euclidean space near each point.
• Topological spaces are the objects studied in topology.
• Topological spaces equipped with extra property and structure form the fundament of much of geometry.
• A topological space equipped with a notion of smooth functions into it is a diffeological space.
• The word ‘topology’ sometimes means the study of topological spaces but here it means the collection of open sets in a topological space.
• Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.

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• ID : Q179899

Spacy 패턴 목록

• [{'LOWER': 'topological'}, {'LEMMA': 'space'}]