Use "Topological Space" in a sentence | "Topological Space" sentence examples

Topological space is a fundamental concept in mathematics that plays a crucial role in various branches of the subject, including topology, analysis, and geometry. Understanding how to use the term "topological space" correctly in sentences is essential for effectively communicating mathematical ideas and concepts. Here are some tips on how to use this exact word or phrase in a sentence:

1. Definition: When introducing the term "topological space" for the first time, it is important to provide a concise and accurate definition.

For example, "A topological space is a set equipped with a collection of subsets, called open sets, that satisfy certain axioms."

2. Examples: To illustrate the concept of a topological space, it is helpful to provide examples. For instance, "The real numbers, equipped with the standard topology, form a topological space." or "The unit circle in the complex plane, with the induced topology, is another example of a topological space."

3. Properties: When discussing topological spaces, it is often necessary to mention their properties. For instance, "In a Hausdorff topological space, any two distinct points can be separated by disjoint open sets." or "A compact topological space is one in which every open cover has a finite subcover."

4. Relationships: Topological spaces can be related to other mathematical concepts.

For example, "A continuous function between two topological spaces preserves the underlying topological structure." or "The notion of a metric space is a special case of a topological space, where the open sets are defined using distances."

5. Applications: It can be helpful to mention the applications of topological spaces in various fields. For instance, "Topological spaces are widely used in computer science for modeling networks and data structures." or "In physics, topological spaces are employed to describe the properties of space-time in general relativity."

6. Historical Context: Providing a brief historical context can add depth to your sentence.

For example, "The concept of a topological space was first introduced by the French mathematician Felix Hausdorff in the early 20th century."

7. Formal Language: When using the term "topological space" in a formal mathematical context, it is important to adhere to the appropriate language and notation.

For example, "Let (X, ?) be a topological space, where X is the underlying set and ? is the collection of open sets." Remember to use the term "topological space" in a way that is consistent with its mathematical definition and context. By following these tips, you can effectively incorporate this term into your mathematical discussions and enhance your understanding of topology.

In the remaining portion of this article, additional example sentences are presented to demonstrate the usage of the term "Topological Space" within sentences.

"Topological Space"

(1) In a topological space

(2) The concept of a topological space is abstract

(3) The infimum of a set can be a limit of a topological space.

(4) The notion of convergence can be defined in a topological space.

(5) The automorphisms of a topological space preserve its open sets.

(6) A topological space can be finite or infinite, discrete or continuous.

(7) The universal property of a topological space defines its continuous maps.

(8) The concept of a topological space is fundamental in the field of topology.

(9) The automorphisms of a topological space can reveal its geometric properties.

(10) A topological space allows us to define notions of continuity and convergence.

Sentence For "Topological Space"

(11) A topological space can be defined by specifying its open sets or closed sets.

(12) The concept of a topological space is closely related to the idea of continuity.

(13) In topology, a topological space is a set equipped with a collection of open sets.

(14) Cohomologies can be used to study the properties of sheaves on a topological space.

(15) The properties of a topological space depend on the collection of open sets chosen.

(16) An automorphism of a topological space is a homeomorphism from the space to itself.

(17) The set of all endomorphisms of a topological space forms a monoid under composition.

(18) The topology of a space refers to the arrangement of open sets in a topological space.

(19) The concept of a topological space generalizes the notion of distance in metric spaces.

(20) The concept of a metric space is closely related to the concept of a topological space.

"Topological Space" In A Sentence

(21) The concept of a topological space is fundamental in the study of topological manifolds.

(22) The concept of a topological space is fundamental in the study of geometry and analysis.

(23) The concept of a topological space is a fundamental building block in modern mathematics.

(24) The open sets in a topological space can be thought of as the building blocks of the space.

(25) The concept of a topological space is closely related to the idea of topological equivalence.

(26) The concept of a topological space is closely related to the idea of neighborhood and proximity.

(27) A topological space can be visualized as a set with a certain arrangement of open and closed sets.

(28) A topological space can be defined by specifying its open sets and the relationships between them.

(29) The automorphism of a topological space can be used to study its homeomorphisms and continuous maps.

(30) The concept of a topological space allows for the study of continuity without the need for a metric.

"Topological Space" Sentence

(31) The definition of a topological space involves specifying the open sets that satisfy certain axioms.

(32) The properties of a topological space can be analyzed using various topological tools and techniques.

(33) The set of all continuous transformations of a topological space forms a semigroup under composition.

(34) Every metric space can be considered as a topological space by defining open sets based on the metric.

(35) The concept of a topological space is essential in understanding the behavior of topological structures.

(36) The concept of a topological space was introduced by mathematician Felix Hausdorff in the early 20th century.

(37) A topological space is a mathematical structure that consists of a set of points and a collection of open sets.

(38) A topological space can have multiple different topologies, each defining a different arrangement of open sets.

(39) The concept of a topological space generalizes the notion of distance and allows for more abstract mathematical analysis.

(40) The concept of a topological space is abstract and general, allowing for the study of a wide range of mathematical objects.

(41) The universal property of a topological space ensures that it is the most general space that satisfies a certain property in topology.

Learning English Faster Through Complete Sentences With "Topological Space"

Sentences are everywhere.
Without sentences, language doesn’t really work.

When you first started learning English, you may have memorized words such as English meaning of the word "Topological Space"; But now that you have a better understanding of the language, there’s a better way for you to learn meaning of "Topological Space" through sentence examples.

True, there are still words that you don’t know. But if you learn whole sentences with "Topological Space", instead of the word "Topological Space" by itself, you can learn a lot faster!

Focus Your English Learning On Sentences With "Topological Space".

Why Is Focusing on Sentences Important?
Sentences are more than just strings of words. They’re thoughts, ideas and stories. Just like letters build words, words build sentences. Sentences build language, and give it personality.

Again, without sentences, there’s no real communication. If you were only reading words right now, you wouldn’t be able to understand what I’m saying to you at all.

- The Word "Topological Space" in Example Sentences.
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- How to use "Topological Space" in a sentence.
- 10 examples of sentences "Topological Space".
- 20 examples of simple sentences "Topological Space".

All the parts of speech in English are used to make sentences. All sentences include two parts: the subject and the verb (this is also known as the predicate). The subject is the person or thing that does something or that is described in the sentence. The verb is the action the person or thing takes or the description of the person or thing. If a sentence doesn’t have a subject and a verb, it is not a complete sentence (e.g., In the sentence “Went to bed,” we don’t know who went to bed).

Four Types Of Sentence Structure.

Simple Sentences With "Topological Space"

A simple sentence with "Topological Space"contains a subject and a verb, and it may also have an object and modifiers. However, it contains only one independent clause.

Compound Sentences With "Topological Space"

A compound sentence with "Topological Space" contains at least two independent clauses. These two independent clauses can be combined with a comma and a coordinating conjunction or with a semicolon.

Complex Sentences With "Topological Space"

A complex sentence with "Topological Space" contains at least one independent clause and at least one dependent clause. Dependent clauses can refer to the subject (who, which) the sequence/time (since, while), or the causal elements (because, if) of the independent clause.

Compound-Complex Sentences With "Topological Space"

Sentence types can also be combined. A compound-complex sentence with "Topological Space" contains at least two independent clauses and at least one dependent clause.