[9up數學] 咩叫維度? (數學角度)

個mapping係有限制的, 例如topologically黎講係homoemorphism (個map要continuous, bijective, inverse都要continuous)
如果個map無限制我地只係單純數cardinality

可唔可以講下 cardinality 同 dimension 嘅關係
我應該喺呢度有啲嘢搞唔清楚

intuitively cardinality係指個set有幾多elements
例如{1, 2, 3, 4, 5}個cardinality係5
但係infinite set的話我地可以比較邊個set "數量多D"
而我地話兩個set cardinality一樣if and only if 佢地之間存在一個bijection

至於dimension, 例如vector space, 咁vector space dimension本身preserved by bijective linear transformation (aka isomorphisms)

homeomorphism就可以preserve topological manifold的dimension

以上兩種by定義, 兩個vector spaces/manifold的cardinality會係一樣

例如 natural numbers 同 integers 同 rational numbers 的cardinality一樣, 但係佢地同real numbers的cardinality唔一樣

會唔會有兩個set A同B
A 嘅 cardinality 大過 B
但 A 嘅 dimension 細過 B

唔可能, intuitively個set多左野自然要用多D野去形容佢

以finite dimensional vector space爲例, 如果 dim A < dim B, 我地可以embed A 落 a subspace of B
e.g. A 的 basis 係 {a_1, a_2, ..., a_m}
B 的basis係 {b_1, b_2, ..., b_m, ..., b_n}
我地可以construct 一個linear transformation f: A -> B such that
f(a_i) = b_i
咁樣就有個injection, 即係cardinality of A最多會係cardinality of B

好的
等我再思考一下

兩個set A同B有無話underlying scalar field要係一樣？

如果base field唔同無得比較喎

Pish