Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Topology and Analysis
Convergence .... Singh, Example 4.1.1 .... .... Another Question ....
Reply to thread
Message
[QUOTE="member 587159, post: 6312847"] ##V=\mathbb{R} \setminus \{x_n: x \neq x_n, n \geq 1\}## is a neighborhood of ##x## as ##V## contains ##x## and it is the complement of a countable set, thus open. But ##x_n \to x##, which means that for all neighborhoods ##U## of ##X##, there is ##n_0## such that ##x_n \in U## if ##n \geq n_0##. Next, take ##U = V##. Then there is ##n_0## such that ##x_n \in V \iff x_n = x## if ##n \geq n_0##. __________________ [B]Addendum for the interested reader:[/B] Note that the converse of what the author claims holds: any sequence that is eventually constant converges (to the eventually constant value). Thus in this topological space, the convergent sequences are exactly these which are eventually constant. On ##\mathbb{R}##, we can also define the discrete topology ##\mathcal{P}(\mathbb{R})## (every subset of ##\mathbb{R}## is open). Then also a sequence in ##\mathbb{R}## is convergent if and only if it is eventually constant (easy exercise). Thus we see that ##(\mathbb{R}, \mathcal{T}_c)## and ##(\mathbb{R}, \mathcal{P}(\mathbb{R}))## are [B]different [/B]topological spaces with the [B]same[/B] convergent sequences. This means that sequences do not suffice to describe the topology (unlike for metric spaces or more generally spaces where every point has a countable neighborhood basis). The solution then is to generalise the concept of sequence and allow more general index sets than ##\mathbb{N}##. We then come to the concept of nets and these do describe the topology. But some authors use other approaches and use filters instead of net, though both concepts are equivalent and can be translated to one another. TLDR: Sequences in topological spaces do not suffice to describe the topology, so there is the need to introduce a new kind of object to describe convergence. This will be a net or a filter. [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Topology and Analysis
Convergence .... Singh, Example 4.1.1 .... .... Another Question ....
Back
Top