Path connectedness, Compactness implies Continuity in a Path connected Complete Metric space
Prabhat Guruprasad Dwivedi Assistant Professor Department of Mathematics Mithibai College
Tal: Vile Parle (W) Dist: Mumbai State: Maharashtra, India
“A continuous function on metric space maps path connected set to path connected set and compact set to compact set.” This is a popular result in general metric space. In this article I prove the converse on path connected complete metric space i.e. “a function on a path connected complete metric space into path connected complete metric space which preserves path connected set and compact set is continuous.”
KEYWORDS: Continuity, Compact set, Connected set, Path Connected set, Metric space.
Introduction:
A continuous function on metric space maps connected set to connected set. But the converse is not true.
Example: Let R denotes set of real numbers in its usual metric.
A function : ⟶ defined as = if ≠ 0 = 0 if = 0
f maps connected set to connected set but f is not continuous at 0.
A continuous function on metric space maps compact set to compact set. But the converse is not true.
Example: Let R denotes set of real numbers in its usual metric.
A function : ⟶ defined as = 1 if ∈ = 0 if ∉
f maps compact set to compact set but f is not continuous R.
I have proved the theorem “A function f : Rm→Rk which maps connected set to connected set and compact set to compact set is continuous.” The results used in proving the theorem were “Any closed and bounded ball in Rn is connected and compact” and
Abstract
In general metric space, closed and bounded ball need not be connected and compact and we cannot say anything about its dimension. Hence we cannot apply the results used in n-Euclidean space to general metric space.
A continuous function on metric space maps path connected set to path connected set.
Every path connected metric space is connected. The converse of this does not hold i.e.
connected metric space need not be path connected. In this article I restricted my study to a function on path connected complete metric space which preserves path connected set and compact set.
Theorem (Main):
Let , and ( , be path connected complete metric space. If : → is a function which maps path connected set to path connected set and compact set to compact set then is continuous function on .
Proof:
Suppose f is not continuous at ∈ .
Then ∃ a sequence of distinct points in such that → and ! , is a strictly decreasing sequence of non negative real numbers which converges to 0 but .
∃ " > 0 such that !$% , & > " > 0 ∀ ∈ ( i.e. ↛
∵ is path connected.
∴ For each ∈ (, ∃ a shortest path between and , i.e. ∃ a continuous function - : . , , / → such that - 0 1 = , - 0 , 1 = , .
∵ . , , / is path connected and compact set.
∴ - . , , / in is path connected and compact set.{ ∵ - is continuous.}
∵ - 0 , 1 = , = - , 0 , 1.
∴ , ∈ - . , , / ∩ - , . ,$, , / .
∴ - . , , / ∩ - , . ,$, , / ≠ ∅.
∴ ⋃ ∈5 - . , , / is path connected set.
Now, ⋃ ∈5. , , /= 0,16.
Define - ∶ [0, 16 → as follows - 0 =
- 9 = - 9 if , ≤ 9 ≤ , for each ∈ (.
We show that - is continuous on [0, 1].
For each ∈ (, -=- on . , , / and hence - is continuous on . , , / and - 0 , 1 = , = - , 0 , 1.
Hence - is continuous on ⋃ ∈5. , , /= 0,16.
Now, we show that - is continuous at 0.
Consider a monotonic decreasing sequence of distinct points 9 in 0,16 converging to 0.
For ∈ (, ∃ ; ∈ ( such that <, ≤ 9 ≤< . But -< is shortest path from < to <, . As → ∞, 9 → 0 and hence ; → ∞.
Also - 9 = -< 9 → <, as 9 →<, which together with → ∞, ; → ∞ implies <, → and hence - 9 → - 0 = .
Thus, 9 → 0 ⟹ - 9 → - 0 = i.e. - is continuous at 0.
Hence - is continuous on [0, 1].
Therefore, - is a path from to and -[0,16 is path connected and compact set.
Consider nested sequence of path connected and compact sets in R and hence its image under - in and then in > under the image of ?- (we use here property of continuous function and
assumption of )
[0,16 ⊃ A0,1
2C ⊃ A0,1
3C ⊃ ⋯ ⊃ A0,1C ⊃ ⋯ -[0,16 ⊃ - A0,1
2C ⊃ - A0,1
3C ⊃ ⋯ ⊃ - A0,1C ⊃ ⋯
?- [0,16 ⊇ ?- A0,1
2C ⊇ ?- A0,1
3C ⊇ ⋯ ⊇ ?- A0,1C ⊇ ⋯ Now, ∈ - [0,16. {∵ γ 0 1 = }
∵ ?- [0,16 is compact set and hence sequential compact set.
∴ A sequence has a convergent subsequence I . Let I converges to y where J ∈ ?- [0,16.
We can easily show that J ∈ ?- .0, / for each ∈ ( and hence J ∈ ⋂ ∈5 ?- .0, / .
∵ ∃ " > 0 such that !$% , & > " > 0 ∀ ∈ (.
∴ ≠ J.
∴ , J ∈ ⋂ ∈5 ?- .0, / .
∴ ⋂ ∈5 ?- .0, / is path connected set containing two distinct points and y hence it is an infinite set and there exist a path between and y.{by the property of path connected sets}
∴ We can construct a sequence of distinct points J in ⋂ ∈5 ?- .0, / with corresponding sequence of distinct points L in - .0, / and 9 in .0, / for each ∈ ( {Note: J = L = ?- 9 } such that J ≠ J and J ≠ but converges to y.
∵ → ∞. ∴ 9 → 0.
∴ L = - 9 → - 0 = . {∵ - is continuous at 0.}
∴ A set {L ∶ ∈ (} ∪ { } is a compact set in X.
But its image set { J = L : ∈ (} ∪ { } is not compact in Y. {as J converges to y and J ≠ J and y≠ . }
Hence we have a contradiction to statement that “ maps compact set to compact set”.
Therefore, our assumption that “ is not continuous at ∈ ’’is wrong.
Thus is continuous at ∈ and on . References:
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