可点缩的空间是否一定以其中某一点作为其强形变收缩核呢？ 第1页

此问题下 @寨森Lambda-CDM 的回答引用了 Hatcher 的 Algebraic Topology 中的一个例子。就此机会，我想提一下他书上关于此问题的另一个例子（事实上就是那道题的后面一题）。个人认为这题比上面那题还坑，所以把自己的解答放在这里以供参考。不得不说，Hatcher 此书中有些构造类问题确实tricky。

我就不翻译了。个别概念和定理引用了书中的内容，如果实在影响理解的话就去翻一下书吧。

7. The "cone on Cantor set" denotes a subset of the unit right triangle , which gives by , where denotes the Cantor set in respect to . And we say in this case is called the vertex of this cone on Cantor set.

denotes the union of infinite repeats of cones on Cantor set, with the baseline , that is, .

Therefore is the closed lower half plane with "fins" attached.

Now consider the one-point compactification space of . We know that the one-point compactification of a closed half plane embeds in as a closed disk (use the stereographic projection). Since the "fins" are already compact themselves, there is no need to compactify them. Therefore the one-point compactification space of embeds in as a closed disk with fins attached. Remark that the fins are still of the same size (it may takes some time to figure out how this works explicitly).

Finally, we attach another cones on Cantor set on by gluing the baseline to the boundary of as a loop. This leads to the desired space .

Now we show that is contractible but not deformation retracts to any single point.

It is not difficult to show that for any point , any neighborhood of is not path-connected, no matter is in the open disk, or on the fins, or on the boundary. Then we know that does not deformation retracts to any point using the conclusion in problem 5 and 6.

To see why is contractible, we use the same method in problem 6, that is to prove deformation retracts in the weak sense to . Firstly, we know that deformation retracts to the vertex of its first cone on Cantor set, i. e. , and for the same reason, deformation retracts to -axis. So we can make all the fins attached to the disk deformation retract onto the disk. Then there remains the cone on Cantor set wrapping on the boundary . As what we did we in problem 6, we can make all the "bristles" retracts to while itself rotates in the same direction and at the same speed, like reeling the fishing line. This gives a deformation retraction in the weak sense to . Therefore, deformation retracts in the weak sense to . Since is contractible, we know that is contractible.

经过搜索，在Hatcher的Chapter0的第6题（具体而言，6(b)）找到了反例

（Hatcher书中的deformation retract的定义其实是很多地方所说的strong deformation retract，也就是题主所说的强形变收缩核。它与可缩的区别就在于，会不会维持那个点不动）

下面简单答一下这两个小问（不严格，直觉式“证明”）：

6(a)：deformation retract到 中的任意点 是显然的，让上面那些“束”降落到 ，然后线段自然可以缩到 ，这个过程连续且维持 不动。但是确实不能retract到其他点（即“束”上的点 ）。如下图，点 旁边的蓝点要想最终跑到点 ，需要先降落到底下的 ，然后到 正下方，再上升到 。由于有理数集是稠密的， 周围有无数个这样的蓝点，可以选取一列蓝点趋于 。但这一列蓝点降落到底部再上升的话， 就不可能不动，否则会破坏连续性。

6(b)：它是可缩的。原始的回答有问题，评论区给出了正确的构造：让所有“束”以同一速度（而不是同一比例）向“折线”收缩，同时所有“折线”上的点以同一速度向右沿“折线”流动。较短的“束”上的点先收缩到“折线”上，然后就随着流动。

但并不能选取某个点作为deformation retract。假如说能选一点 作为deformation retract的话，根据6(a)， 只能落在中间蜿蜒的折线上，如下图所示。但是总可以仿照刚才选取周围的一列蓝点趋于 ，这些蓝点降落到蜿蜒的折线上再折回 会迫使点 运动，否则破坏连续性。