continuity – Prove that $ mathbb {N} $ is homeomorphic to $ {(1 / n): n ge 1} $

Prove it $ mathbb {N} $ (with the usual metric) is homeomorphic to $ {(1 / n): number 1 } $ (with usual metric)

I have defined a function $ f: n rightarrow 1 / n $ which is clearly one-on and on. But I can not prove that $ f $ And $ f ^ {- 1} $ is continuous in the metric space.

Thank you!

Geometry dg.differential – In a variety, $ angle xpy> frac { pi} {2} $, for $ q $ on $ px $ or $ py $, $ B_q (r) $ homeomorphic to $ B_p (r ) $?

Let M be a Riemannian variety with n dimensions, without border, with sectional curvature $ geqslant -1 $. For one point $ p in M ​​$, suppose that there is $ l, delta> $ 0, $ x, y $ M with $ d (p, x), d (p, y)> l $ and a geodesic $ px $ and $ py $ with angle $ angle xpy> frac { pi} {2} + delta $. Let $ q $ to be a geodesic point $ px $ or $ py $, Question: is there $ r> $ 0, which depends only on $ n, l, delta $ such as $ B_q (r) $ is homeomorphic to $ B_p (r) $?

Equally, we can ask the question as follows:

Let $ M_i $ to be a sequence of Riemannian varieties with $ sec geqslant -1 $ and diameter $ leqslant $. assume $ (M_i, p_i) $ Gromov-Hausdorff converge (eventually collapse) towards $ (X, p) $ (we know that this is an Alexandrov space). Suppose that there is $ l> 0, delta> $ 0, $ x, y in X $ with $ angle xpy> frac {pi} {2} + delta $. elevator $ x, y $ at $ M_i $, we have $ x_i, y_i in M_i $. with$ angle x_i p_i y_i> frac { pi} {2} + delta $. Let $ q_i $ to be a geodesic point $ p_ix_i $ or $ p_iy_i $. Question: Is there $ r> $ 0, such as $ B_ {q_i} (r) $ is homeomorphic to $ B_ {p_i} (r) $?

general topology – Show that $ widehat {(0,1)} $ is homeomorphic to $[0,1]/ sim $ where $ x sim y iff x = y text {or} {x, y } = {0,1 } $.

$ widehat {(0,1)} = (0,1) cup {p } $, or $ p not in (0,1) $. A set $ U $ in $ widehat {(0,1)} $ is open if ($ U not nor p $ and $ U $ is open in $ (0.1) $) or ($ U ni p $ and $ (0,1) setminus U $ is a compact subset of $ (0.1) $).

CA watch $ widehat {(0,1)} $ is homeomorphic to $[0,1]/ sim $ or $ x sim y ssi x = y text {or} {x, y } = {0,1 } $.

A set of representatives of $[0,1]/ sim = {[x] mid x in[0,1] } $ is $ bigcup_ {x in (0,1)} underbrace {[x]} _ {= {x }} cup underbrace {[0]} _ {=[1]= {0,1 }} $.

Let $ p:[0,1] twoheadrightarrow [0,1]/ sim: x mapsto [x]$ to be the quotient map, so $ U subset[0,1]/ sim $ is open if $ p ^ {- 1} (U) subset [0,1]$ is open.

I have to prove that
$$ begin {align *} f colon[0,1]/ sim & longrightarrow widehat {(0,1)} \ [x]& longmapsto x text {if} x in (0,1) \ [0]=[1]& longmapsto p end {align *} $$

is a homeomorphism.

It is clear that it is a bijection. I have problems to prove that it is continuous.

Let $ U subset widehat {(0,1)} $ open. assume $ p not in U $then $ U $ is open in $ (0.1) $, So $ U = (0,1) cap B $ ($ B $ open in $ mathbf {R} $). then $ f ^ {- 1} (U) $ is open in $[0,1]/ sim $, since $ p ^ {- 1} (f ^ {- 1} (U)) = U $which is open in $[0,1]$, since $ U = underbrace {((0,1) cap B)} _ { text {open in} mathbf {R}} cap [0,1]$.

Now assume $ to U $. then $ (0,1) setminus U $ is a compact subset of $ (0.1) $ (I guess that means by Heine Borel that he's closed $ (0.1) $?) So $ (0,1) setminus U = (0,1) cap C $ ($ C $ shut in $ mathbf {R} $). Now $ (0,1) setminus ((0,1) setminus U) = (0,1) cap underbrace {( mathbf {R} setminus C)} _ { text {open in} mathbf {R}} $. It means that $ p ^ {- 1} (f ^ {- 1} (U)) $ is the union of an open subset of $ (0.1) $ and $ {0,1 } $, which is not open in $[0,1]$.

I think this problem is difficult for me because I do not feel familiar with the topology of the quotient yet. Could someone provide some help or a simpler way to handle this?

Topology gt.geometric – Are triangulations with common refinements homeomorphic?

Are there simple triangulations $ K_1 $ and $ K_2 $ of a topological variety $ M $ such as $ K_1 $ and $ K_2 $ have a common subdivision but they are not homeomorphic PL? Ideally, I would like an example in dimension 4.

From what I understand from PL homeomorphism, there is a common subdivision $ L $ and embarkations $ phi_i $ of $ K_i $ to a Euclidean space that takes simplexes of both $ K_i $ and of $ L $ to linear simplexes. Note that there is a triangulation $ L $ 3 simplex K $ which contains clover in its 1-skeleton with only 3 edges. As the number of clover sticks is 6, there is no incorporation $ phi: K to mathbb {R} ^ n $ which is linear on both simplexes of K $ and $ L $.

Any suggested references would also be welcome.

General Topology – The punctured balloon is not homeomorphic to the Euclidean space

Just another ordinary day with another (big) ordinary math conversation. A friend and I asked this:

Problem. Prove it $ B (0, r) setminus {0 } subseteq mathbb {R} ^ n $ is not homeomorphic to open bullets for $ r> $ 0.

I have not taken a topology course yet and my friend has just started a topology course. So we tried to find as basic a solution as possible.

It seems easy enough, even though we struggled …

  • It is enough to prove that it is not homeomorphic $ mathbb {R} ^ n $.
  • Most elemental topological invariants do not work.

The best we have found is to compute the fundamental group and take generalizations of the fundamental group for $ n> $ 2. This should probably work.

So here's my question: Is there another way to prove this result?

Algebraic topology – Are the level sets of a smooth map of a topological space homeomorphic to a collection of cells?

This question is related to the definition of Euler-Integral transformations, page 147 of Michael Robinson's book "Topological signal processing".

Definition: characteristic of Euler and constructable function

The euler characteristic $ chi $ is a valuation of a cell complex $ X_f $ at $ mathbb {Z} $ Defined by:
$$ chi (X_f) = sum_ {c in X_f} ^ {} {- 1 ^ {dim (c)}} $$

$ { bf Note:} $The definition of $ chi $ allows to define it for an arbitrary collection of cells. For example, a sub-collection $ X_f $ of cells in $ X_f $ is not always a cellular complex, but we will admit that $ chi (X_f & # 39;) $ is well defined. Even though the author does not say so clearly in the book, this hypothesis seems to be widely used.

A whole-valued constructible function is a function of a topological space $ f: X mapsto mathbb {Z} $ such that there is a non-unique cellular complex $ X_f $ For who:

a) There is a homeomorphism $ h: X mapsto X_f $

b) the function $ f circ h $ is constant on every cell of $ X_f $

This can be seen as a generalization of the piecewise constant function on topological spaces having a complex cellular skeleton.


Statement of the problem

Given a topological space $ X $ and a smooth map $ P: X mapsto mathbb {R} $. We suppose that $ X $ is such that there is a homeomorphism $ h $ enter $ X $ and a cellular complex $ X_f $ . We define the sets of levels of $ P $:
$$ X_c = {x in X, P_x (x, bullet) = s } $$

The question is: can we calculate $ chi (h (X_c)) $ ?

The author seems to do it. But I am troubled by the fact that the Euler feature is only defined for cell collection and that the set defined above might very well not be a collection of cells.


For example, if $ Y $ is the ball centered in o in $ mathbb {R} ^ 2 $ with the complex cellular structure made of a $ 0-cell, a $ 1 $-cell and a $ 2 $-cell. Let us $ P: y mapsto | y | $.

The level set of $ P $ for $ s $ are the intersection between a circle of radius s and $ Y $. Three cases can occur:

  1. For $ s = $ 0, the defined level is the single point (0,0).
  2. For $ 0 <s <{max} $ Y_s is a closed circle inside $ Y $.
  3. For $ s = s_ {max} $ the level defined is the border of the $ 1 $-cell attached to the $ 0-cell.

It seems that apart from the third case, the defined levels are neither cells nor a set of cells. This explains my inability to understand how the author can apply the characteristic euler on these sets.

I wonder if anything has escaped me in $ h $ to be a homeomorphism, or by $ P $ to be smooth, as I do not seem to use any of these assumptions. I would appreciate some clues as to what the author actually tries to do (for those who read the book) or on the accuracy of the Euler features on these sets.

Two complex manifolds homeomorphic non diffeomorphic

Is there a closed topological variety supporting two non-diffeomorphous smooth structures, both of which have a complex, compatible structure? Same question, but for the symplectic structure.