Tagged Questions
8
votes
1answer
82 views
Topological manifolds (dimension)
I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
-2
votes
2answers
37 views
$T:V\rightarrow W$ such that $R(T) \subset W'$ is a subspace of ${\cal{L}}(V,W)$ [closed]
Let $V$ and $W$ be finite-dimensional vector spaces over $F$ and $W'\subset W$ a subspace, then the subset ${\cal{L}}(V,W)$ consisting of all linear maps $T:V\rightarrow W$ such that $R(T) \subset W'$ ...
0
votes
1answer
30 views
Codimensionality: On Cardinality of Linear Equations
How does the codimension of a subspace give the number of linear equations needed to define the subspace?
4
votes
1answer
51 views
Dimension of subspace is greater then dimension of space
If $X$ is a topological space then it's (covering) dimension is defined as a minimal number $n$ such that for every finite open cover $\{U\}$ of $X$ there is a finite open cover $\{V\}$ of $X$ that ...
0
votes
0answers
43 views
Separation of Euclidean Space
Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
14
votes
5answers
204 views
Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples
In lecture 4 of his Introduction to Dynamical Linear Systems course, right after interpreting the inner product in ${\mathbb R}^N$ in terms of the cosine of the subtended angle, Stanford's Stephen ...
7
votes
1answer
358 views
For an $n$-dimensional object, how many types of holes are possible?
Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With ...
1
vote
2answers
126 views
Dimension of a subset
For a closed subset $Y$ of a space $X$ we have the following inequlity of topological (covering) dimensions: $$\dim{Y} \leq \dim{X}$$ (assuming at least one of those is finite).
I have two questions ...
