# Category:Continuous Mappings

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This category contains results about **continuous mappings** in the context of **Topology**.

## Subcategories

This category has the following 20 subcategories, out of 20 total.

## Pages in category "Continuous Mappings"

The following 117 pages are in this category, out of 117 total.

### C

- Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous
- Closed Image of Closure of Set under Continuous Mapping equals Closure of Image
- Closure of Image under Continuous Mapping is not necessarily Image of Closure
- Closure of Preimage under Continuous Mapping is not necessarily Preimage of Closure
- Combination Theorem for Continuous Mappings
- Compactness is Preserved under Continuous Surjection
- Compactness Properties Preserved under Continuous Surjection
- Compactness Properties Preserved under Projection Mapping
- Complex-Differentiable Function is Continuous
- Composite of Continuous Mappings is Continuous
- Composite of Continuous Mappings is Continuous/Point
- Composite of Continuous Mappings on Metric Spaces is Continuous
- Constant Function is Continuous
- Constant Mapping is Continuous
- Continuity Defined by Closure
- Continuity Defined from Closed Sets
- Continuity in Initial Topology
- Continuity of Composite with Inclusion
- Continuity of Linear Transformations/Normed Vector Space
- Continuity of Mapping between Metric Spaces by Closed Sets
- Continuity Test for Real-Valued Functions
- Continuity Test using Basis
- Continuity Test using Sub-Basis
- Continuous Bijection from Compact to Hausdorff is Homeomorphism
- Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary
- Continuous Image of Connected Space is Connected/Corollary 2
- Continuous Involution is Homeomorphism
- Continuous Linear Transformations form Subspace of Linear Transformations
- Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness
- Continuous Mapping is Continuous on Induced Topological Spaces
- Continuous Mapping is Measurable
- Continuous Mapping is Sequentially Continuous
- Continuous Mapping is Sequentially Continuous/Corollary
- Continuous Mapping of Separation
- Continuous Mapping on Finite Union of Closed Sets
- Continuous Mapping on Union of Open Sets
- Continuous Mapping to Product Space
- Continuous Mapping to Product Space/Corollary
- Continuous Mapping to Product Space/General Result
- Continuous Mappings preserve Convergent Sequences
- Countability Axioms Preserved under Open Continuous Surjection
- Countability Properties Preserved under Projection Mapping
- Countable Compactness is Preserved under Continuous Surjection

### D

- Derivative Operator on Continuously Differentiable Function Space with C^1 Norm is Continuous
- Derivative Operator on Continuously Differentiable Function Space with Supremum Norm is not Continuous
- Diagonal Operator over 2-Sequence Space is Continuous Linear Transformation
- Distance from Point to Subset is Continuous Function
- Distance Function of Metric Space is Continuous
- Domain of Continuous Injection to Hausdorff Space is Hausdorff

### E

### F

### G

### I

### L

- Left Shift Operator on 2-Sequence Space is Continuous
- Lindelöf Property is Preserved under Continuous Surjection
- Linear Integral Bounded Operator is Continuous
- Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous
- Local Connectedness is not Preserved under Continuous Mapping

### M

- Mapping from Cartesian Product under Chebyshev Distance to Real Number Line is Continuous
- Mapping from L1 Space to Real Number Space is Continuous
- Mapping from Standard Discrete Metric on Real Number Line is Continuous
- Mapping whose Graph is Closed in Chebyshev Product is not necessarily Continuous
- Definition:Mappings Separating Points
- Definition:Mappings Separating Points from Closed Sets
- Maximum Rule for Continuous Functions
- Metric is Continous Mapping
- Minimum Rule for Continuous Functions

### P

- Paracompactness is not always Preserved under Open Continuous Mapping
- Paracompactness is Preserved under Projections
- Pasting Lemma
- Pasting Lemma for Continuous Mappings on Closed Sets
- Pasting Lemma for Continuous Mappings on Open Sets
- Pasting Lemma for Pair of Continuous Mappings on Closed Sets
- Pasting Lemma for Pair of Continuous Mappings on Open Sets
- Pasting Lemma/Continuous Mappings on Closed Sets
- Pasting Lemma/Continuous Mappings on Open Sets
- Pasting Lemma/Counterexample of Infinite Union of Closed Sets
- Pasting Lemma/Finite Union of Closed Sets
- Pasting Lemma/Union of Open Sets
- Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets
- Projection from Product Topology is Continuous
- Projection from Product Topology is Open and Continuous

### R

### S

- Second-Countability is Preserved under Open Continuous Surjection
- Sequential Compactness is Preserved under Continuous Surjection
- Sigma-Compactness is Preserved under Continuous Surjection
- Supremum Operator Norm as Universal Upper Bound
- Supremum Operator Norm is Norm
- Supremum Operator Norm is Well-Defined